Fun with Coins: Optimizing The OR Cafe Price Menu

We've looked at a few coin-optimization models here before. ... We'll look at another one today. We'll try to price the menu at OR cafe. You can look at the detailed problem statement and the associated sample data here. As always, there's no claim that the method described here is a 'best' or even a 'correct' way to manage such a problem. This post serves as a useful starting point for analyzing this fun problem with coins and change.

Basically, the problem we are analyzing in this post here is to change the existing prices of items in the menu by tiny amounts (+/- 2 cents) to maximize the number of (future) customer bills that end with zero or five cents (i.e., $x.05, $x.10, $x.15, ... $x.95) after (13%) tax. Assuming that past transactions (200 of them) are a good indicator of future purchase patterns, we can select a portion of this data (say 60%) to 'train' our pricing model (MIP) on, and then evaluate its performance on the remaining 40% of the transactions.

Model
1. Decision variables: There are five pricing possibilities for the recommended price of an item p(i) = {p0(i)-2, p0(i)-1, p0(i), p0(i)+1, p0(i)+2}, where p0(i) = original price of the item in the menu.

2. Bill(transaction j) b(j) = sum(i) Aij.p(i)*(1+T/100)
where:
a. Aij = number of units of item (i) purchased in historical transaction (j)
b. T = percentage sales tax charged

3.The minimum (l(j)) and maximum (u(j)) values for b can be determined by setting all prices to their lower bounds, and upper bounds respectively. We assume that fractional cents are rounded to the nearest cent. This gives a pre-computable range of feasible b values V(j) = {l(j), l(j)+1, ...., u(j)}.

4. Objective: Maximize the number of times b ends in 0 or 5 (using boolean indicator variable 'z' and 0-1 cost 'c') in the training set by selecting one price from the 5 possibilities for each item

CAFEMIP: Max sum(k,j) c(k,j)z(k,j)
subject to:
ii)  sum(i) Aij.p(i)*(1+T/100) = sum(k)V(k, j).z(k, j) for all j
iii) SOS-1 constraints for: prices p(i) that ensure that one of five prices is picked for each item in the menu, and for z(j) to ensure that exactly one bill realization is active for each transaction.

Round-off Issue
The LHS of constraint 4(ii) can contain fractional cents, whereas the RHS only admits discrete values. This situation can be overcome as follows:
b(j) = sum(i) Aij.p(i) + sum(i) Aij.p(i)(T/100) = b'(j) + tax,
where b'(j) is integer. Every realization of b'(j) (=V'(j)) can be mapped to our 0-1 objective function contribution after accounting for the additional tax it generates. Thus, it is sufficient to retain the integer components of 4(ii) and rewrite it free of round-off error as:

sum(i) Aij.p(i) = sum(k)V'(k, j).z(k, j) for all j

CAFEMIP was optimized using CPLEX 12.x and implemented in Java. As a secondary objective, a tiny penalty was added to minimize the deviation of prices from the original value to avoid unnecessary price changes or revenue shifts.

Results
1. optimized over the entire 200 transactions (no hold-out sample).We count the number of times, the cafe needed to round to the nearest 0 or 5 using the prices before and after optimization.

Roundings needed: original price:   135/200
Roundings needed: optimized price: 35/200
While the results on training data are really good, we do not know how well CAFEMIP would do on hidden data, so we re-run CPLEX on a subset of the data.

2. Optimized over the first 120 transactions only, and tested on the last (hidden) 80 transactions.
training data: original  cost =     77/120
training data: optimized cost = 20/120

--- CPLEX log ----

Tried aggregator 1 time.
MIP Presolve eliminated 34 rows and 17 columns.
Reduced MIP has 230 rows, 847 columns, and 1824 nonzeros.
Reduced MIP has 835 binaries, 0 generals, 0 SOSs, and 0 indicators.
Probing fixed 68 vars, tightened 0 bounds.
Probing time =    0.01 sec.
Tried aggregator 1 time.
MIP Presolve eliminated 4 rows and 207 columns.
MIP Presolve modified 14 coefficients.
Reduced MIP has 226 rows, 640 columns, and 1666 nonzeros.
Reduced MIP has 629 binaries, 11 generals, 0 SOSs, and 0 indicators.
Presolve time =    0.06 sec.
Probing time =    0.02 sec.
Clique table members: 3071.
MIP emphasis: balance optimality and feasibility.
MIP search method: dynamic search.
Parallel mode: deterministic, using up to 4 threads.
Root relaxation solution time =    0.00 sec.

        Nodes                                         Cuts/
   Node  Left     Objective  IInf  Best Integer     Best Node    ItCnt     Gap

      0     0       13.3095    48                     13.3095       92         
*     0+    0                           73.0000       13.3095       92   81.77%
      0     0       18.8333    13       73.0000     Cuts: 130      189   74.20%
      0     0       20.0000     2       73.0000      Cuts: 22      220   72.60%
*     0+    0                           20.0000       20.0000      220    0.00%
      0     0        cutoff             20.0000       20.0000      220    0.00%
Elapsed real time =   0.16 sec. (tree size =  0.00 MB, solutions = 2)

Clique cuts applied:  34
Cover cuts applied:  35
Implied bound cuts applied:  4
Mixed integer rounding cuts applied:  7
Zero-half cuts applied:  7
Gomory fractional cuts applied:  28

Root node processing (before b&c):
  Real time             =    0.09
Parallel b&c, 4 threads:
  Real time             =    0.00
  Sync time (average)   =    0.00
  Wait time (average)   =    0.00
                          -------
Total (root+branch&cut) =    0.09 sec.
CPLEX objval = 20.000000000000025

original, new price of XS Coffee is $1.24, $1.24
original, new price of S Coffee is   $1.33, $1.33
original, new price of M Coffee is   $1.57, $1.55
original, new price of L Coffee is    $1.71, $1.73
original, new price of XL Coffee is  $1.91, $1.90
original, new price of S Tea is        $1.33, $1.32
original, new price of M Tea is       $1.52, $1.51
original, new price of L Tea is        $1.71, $1.73
original, new price of Donut is       $0.95, $0.97
original, new price of Bagel is        $1.24, $1.24
original, new price of Muffin is       $1.30, $1.28
original, new price of Cookie is      $0.75, $0.75


(prices in red indicate a changed recommendation when optimizing over the partial sample instead of the entire history)

These prices were used to recompute the bills in the hidden sample:
hidden data: original  cost =   58/80 (~72%)
hidden data: optimized cost = 17/80 (~21%)
Thus, we would have been able to significantly reduce the number of roundings on the hidden sample if we used the CAFEMIP prices instead of the original values. If the customer buying pattern remains the same, we can expect these prices to continue to do a good job. 

Optimally Managing Ladybugs

My daughter loves ladybugs.  However, when it comes to cleverness, many have noted how bees and ants (which seem to have a special place in Indian hearts :) "solve" traveling salesman and shortest path instances that matter to them (see this interesting link as well). In contrast, a ladybug's decision analytics appears to be somewhat shaky. They don't seem to have the kind of auto-GPS of bees and ants (at least in the context of this post), and seem to get confused and stray from their optimal path.

(pic source)

Recently, a number of LBs invaded my residence, and I spent a lot of time trying to shoo them away and determine how they got inside in the first place so that I could block their entry. Then, I came across this website. Apparently, ladybugs prefer humidity and warmth and try to enter homes as winter begins. OK. However, a good number also enter the house come spring time, which was puzzling. It seems some of the LBs goof up. Ladybugs roosting under windows outside the house, with a certain probability, enter the house instead of enjoying spring time outside. Once inside, they suffer from dehydration and die (unless they figure out the route to the bathroom or kitchen). If we try to force them out, they get stressed and shed 'yellow blood' that can stain walls. The optimal ladybug solution for this time of the year is to simply open the windows, so the ones who came inside by mistake can leave. It's a win-win for both parties, and it actually worked. On the other hand, during winter, we have to try and provide a cozy, alternative home in backyard and insulate the house well to keep them out.

The "Chaos" of India

Thanks to the person who pointed me to an interesting article in some online magazine called "livemint" : "Why Rahul Gandhi has understood the chaos of India"

As the 2014 General Election in India approaches, and pro-establishment journos crank out essays defending their favorite candidates, the author of the above article attempts something truly brave: resurrect a universally panned speech by a leading politician of the incumbent ruling political party. In particular, this paragraph caught my attention:
"... There are two things that stand out in the speech. The first is Rahul’s recognition of the chaos of India, for which he used the image of the beehive. I think this is wonderful realism, and excellent imagery."

As a keen India observer and commentator pointed out on twitter: "....beehive is nt chaotic"

Is India really chaotic?
To many a visitor from the west, it has appeared so. A vast section of India's English media who have worshiped at the altar of Western Universalism and return to lecture India's unwashed masses, may readily concur.

India, for a long, long time, has functioned as a highly decentralized system - in virtually every aspect of life. It's workforce contains among the highest percentage of entrepreneurs in the world, and traditionally possesses a remarkably decentralized economic model. It's indigenous religions are decentralized and base on empirical self-realization. There is no single holy book or authority (there is an entire reference library, with scope for new books to be written). However, this lack of a central authority does not imply chaos or that India was never a nation. It is more likely that what is perceived as chaos is in fact a cognitive overload on the part of the casual observer - as noted in Rajiv Malhotra's book "Being Different". The best example for this apparent chaos that actually produces incredibly effective results is the Kumbh Mela (the largest human gathering on earth, which has been recurring for thousands of years now) that recently concluded in India. Several observers from Harvard and other big-brand western institutions showed up this time to analyze this amazing event. Here's what Rajiv Malhotra says about the Mela:

"....India's Kumbha Mela amply demonstrates that diversity can be self-organized and not anarchic, even on a very large scale. Held every twelve years, this is the world's largest gathering of people, attracting tens of millions of individuals from all corners of India, from all strata of society, and from all kinds of traditions, ethnicities and languages. Yet there is no central organizing body, no 'event manager' to send out invitations or draw up a schedule, nobody in charge to promote it, no centralized registration system to get admitted. Nobody has official authority or ownership of the event, which is spontaneous and 'belongs' to the public domain. Since time immemorial, numerous groups have put up their own mini-townships and millions go as individuals just to participate in the festivities."

How and Why? Rajiv argues in his introduction to the chapter "Order and Chaos":
"Indians tend to be more relaxed in unpredictable situations than westerners. Indians indeed find it natural to engage in non-linear thinking, juxtaposing opposites and tackling complexities that cannot be reduced to simple concepts or terms. They may be said even to thrive on ambiguity, doubt, uncertainty, multitasking, and in the absence of centralized authority and normative codes. .... In the vast canon of classical writings in Sanskrit, we see many context-sensitive and flexible ways of dealing with chaos and difference. The search here is always for balance and equilibrium with the 'rights' of chaos acknowledged. On the other hand, in the creation stories in Genesis and in the Greek classics, there is a constant zero-sum battle between the two poles in which order must triumph."

Interestingly, the socio-biological systems of beehives and ant colonies, which may appear chaotic to the casual observer has an underlying order. In his brilliant book 'Traffic: why we drive the way we do', Tom Vanderbilt quotes studies that note how vast numbers of ants move without colliding (without a centralized traffic cop ant), which is certainly worthy of emulation by human car drivers. The author notes "looking at the swarm as a whole, one might not see what is driving the movement". The insects have their own Kumbha Mela going, perhaps.

Rahul Gandhi's 'beehive' metaphor may not be unreasonable, but it appears to have been chosen based on an entirely shallow reasoning due to his inability to decipher India's apparent "complexity", which he contrasts with China's 'simplicity'. India can be plausibly compared in certain contexts to a beehive or an ant colony not because of 'chaotic frenzied activity', but because of an certain well-functioning harmonic order underlying the apparent chaos.

The lesson here for those of us wishing to properly study India is to look beyond the obvious, look deeper, pay attention to context, and there are amazing discoveries waiting to be made. When Rahul Gandhi, the 4th edition of the Nehru dynasty and potential contender for the post of prime-ministership in 2014 compares India to a beehive for all the wrong reasons; when sections of the Indian English media jump in to spin this into some 'deep thought', it only betrays a fundamental lack of understanding of the essence of India: pluralism, diversity, and cooperative decentralization, blissfully unaware that at its deepest philosophical levels, dharmic India is bound by what Rajiv Malhotra terms an 'Integral Unity'.

Synchronized Scheduling Case Study - Part 3

The case study is introduced here.
The ICAR goal is derived here.

Model
The ICAR problem can be specified as follows:
Find a set of feasible crew pairings and aircraft strings such that:
a. every flight in the LEG schedule is covered exactly once by an aircraft string

b. every flight in the LEG schedule is covered exactly once by a crew pairing

c. any string and pairing is used no more than once (let's denote the non-zero pairings and strings as 'active')

d. the total cost is minimized

e. if an active crew pairing contains a connection t(i→j) in T, and there is also some active aircraft string that also contains t, then we say that the crew turns with the aircraft and is thus synchronized. Downstream systems can ensure that the actual crew assigned to flights i and j remain with the actual aircraft tail number that flies i and then j.

f. Maximally satisfy a reliability measure that is proportional to the tightness and the number of the synchronized connections in T. For the purpose of illustration in this post, we specify that a certain minimum count (m) of the connections in T be synchronized. The aim here is to be smart about picking the right airports and connections to synchronize to boost reliability as well as utilization.

g. The active strings belonging to a sub-fleet should form a Hamiltonian circuit. This is a maintenance requirement that tries to ensure equal wear-and-tear across the fleet. There are other constraints that are also interesting topics in their own right, but we'll ignore them here for brevity.

Resource Synchronization Modeling
Our main focus here is on requirement (f). We provide some sample approaches to handling synchronization in large-scale scheduling:
a) Within the subproblem: 
Really tight day-connections t in T can be directly enforced by restricting t to be the only valid connection out of i and into j (i→j) in the flight network. Thus any pairing or string that comes of the subproblem that covers flight i will subsequently cover j. However, we cannot apply this  idea to all connections in T since that is likely to be overly restrictive. 

b) Within the Master program: 
Let p(t)  = sum of all pairing variables that contain connection t
q(t)  = sum of all string variables that contain connection t
Note that p and q will be no more than 1.0. We can impose the following synchronization constraint: 
∑_T p(t)q(t) ≥ m 



The bilinear terms in the reliability constraint can be linearized in a straightforward manner. A personal favorite is the reformulation-linearization technique (RLT). z(t) ≡ p(t)q(t) ⇒

∑_T z(t) ≥ m, z(t) ≤ p(t), z(t) ≤ q(t), z(t) ≥ p(t) + q(t) - 1

Thus, by adding O(T) very sparse constraints and continuous variables z, we can model the synchronization component of ICAR without requiring any approximation, and the resulting formulation remains tractable. The RLT can be applied in a similar manner to synchronize three or more resources as well. The keys for this 'frontal assault' to work are:
1. The cardinality of set T be manageable in the context of the application.
2. Synchronizing two resources is easier than juggling three or more. We can see from the synchronization constraint that if the p-variables are fixed, the z-variables can be eliminated, leaving us with just the q-variables. This 'two variable per constraint' structure makes life much easier during column generation. The synchronization duals associated with the connections in T tend to work much more effectively in generating string-pairing pairs that cover a beneficial subset of common turns in T.

Epilogue

The ICAR solutions that were actually implemented (different from the illustrative version presented here) significantly improved upon the existing approach. It reduced business cycle time, greatly increased reliability, and reduced costs. The success of the ideas employed to tackle ICAR spurred the development of new, algorithmic techniques to solve the next-generation integrated crew scheduling problems that went on to save the LEG carrier many million bottom-line dollars annually during some difficult economic times. As far as the competition with LCA was concerned (for which this solution was designed): LEG had incurred enough losses and pulled the plug. Operations Research successful, patient dead.

Synchronized Scheduling Case Study - Part 2

The case study was introduced in the previous post. Read the introduction here.

.... Despite their suboptimal approach, the schedulers of the LEG carrier did some nice work with their synchronization approach.

2SAT versus 3SAT
There is a jump in decision complexity when moving from a two-body conflict resolution problem to three-body conflict-resolution. Intuitively, we can observe this in linear programming (LP) problems. LPs limited to two variables per constraint are easier to solve. The dual (two constraints per variable) is a generalized network flow problem that can be solved using purely combinatorial methods. Intuitively, one can adopt an alternating heuristic that iteratively fixes one variable and optimizes the resulting univariate problem. In computational complexity theory, SAT-2 is easier to manage compared to SAT-3.

Rather than trying to simultaneously juggle flight attendant (FA), pilot, and aircraft maintenance schedules, the LEG schedulers adopted an simplification. They combined the two human elements into a single 'crew' scheduling entity. This can be achieved via "co-pairing", i.e., by building common pairings (partial schedules) for FAs and pilots employing X = {X_PILOT ∪ X_FA}, i.e., every pairing would be feasible to both pilot and FA contracts and FAA regulations. Column generation allows you incredible flexibility in accommodating such a change. This flexibility is invaluable when hit by a perfect storm. Now, they could work with a two-variable per constraint structure so that their telephonic "alternating heuristic" negotiations could converge in a timely manner. Unfortunately, such a beneficial reduction in complexity was accompanied by inefficiency. The approach erred inordinately on the conservative side. The resources were now too tightly integrated, resulting in a cost spike due to under-utilization of expensive resources. LEG's R&D team was invited to resolve this problem.

Integrated Crew and Aircraft Routing (ICAR)
The OR team quite naturally proposed an integrated scheduling approach that would reduce costs, improve reliability of operations, and also enable tactical synchronization of flight connections. However, the number of possible connections to be evaluated is O(n²) for a n-flight schedule. Methods in the literature are typically data-agnostic; sophisticated alternatives to the straightforward approach designed to perform an acceptable job even in the worst case. Luckily, LEG's OR team had to solve only LEG's problem and not that of every future airline and space shuttle.

The Beauty and the Brute Force
Given the problem size spread over a five-day period, there are a few million candidates to synchronize in the theoretical worst case. However, our focus is only on same and overnight connections, which reduces the candidate list to a few hundred thousand. When we recognize that we only have to focus on the set T of connections of the same sub-fleet type at the same airport (duh), and exploiting a few other contextual restrictions, this final number is data-driven. It turned out that there were no more than tens of thousands of synchronization possibilities to deal with, 99% less than the theoretical bound. This finding allowed the possibility of cheerfully discarding (as we found out a couple of years later when some of the first journal papers on ICAR came out) all the neat out-flanking theorems and approximations presented in the literature.  The method chosen may score low on aesthetics but was effective and flexible in managing the full blown ICAR instances in context. There is a childlike joy in discovering that enumeration solves your practical business problem - akin to finding out that this emperor has very little clothes. This basic counting step gave the team a clear goal.

Problem Statement (at last)
Given a flight schedule, jointly construct feasible crew pairings and aircraft strings such that their scheduling intersection (elements within the connection set T) are optimally synchronized.

And, yes, there were a few TSP (Traveling Salesman Problem) instances to be managed within ICAR. To be continued....

Synchronized Scheduling Case Study - Part 1

Synopsis 
When a low-cost airline (LCA) entered a short-haul aviation market many, many years ago, a legacy airline carrier (LEG) decided to compete with LCA on its own turf by matching the LCA's fare, frequency and super-quick aircraft turn-around times on ground ('turns' for short). This required a high degree of coordination between LEG's in-flight crews (pilots and flight-attendants), ground-crew, airplane maintenance planners, and the airline schedulers (including their scheduling optimization system). In particular, the utilization of the pilots and the flight-attendants had to be maximized. The challenge however was that all these different resources were part of separate organizational structures and were governed by different contractual rules, and the LEG idea eventually turned out to be cost-prohibitive and unreliable due to poor synchronization of resources.  On the other hand, the Operations Research (O. R) team built a specialized scheduling system for LEG (far ahead of the literature available at that time) that would've achieved the needful. Unfortunately, the silo-ed organizational structure hindered adoption.  Result:  Unlike the legendary revenue optimization contest where the legacy carrier triumphed, the tables were turned here, and the LCA flourished. The low-cost LEG idea was resuscitated many years later in a different context and market, and did a better job using the lessons learned from this early setback. In the next few posts, we will compare the two approaches used to synchronize these resource scheduling tasks and summarize the lessons learned.

Airline Scheduling Basics
Given an aircraft schedule, the resource planning is done sequentially in order of their profitability (or cost) to the airline. Thus:
LEG flight schedule → pilot schedule→ flight attendant schedule → aircraft maintenance routing

The cost-critical task here is the construction of "pairings" (or "strings" for aircraft), which denote partial (unassigned) solutions that represents partial tours of a resource. For LEG pilots (set of two persons), and FAs (set of three persons), this represents a tour in a time-space network that starts from a crew domicile (a small subset of airports in the LAC schedule), traverses a bunch of different airports over 1-5 duty days with intervening periods of rest known as layovers, before returning to the home domicile, while satisfying a myriad of contractual and FAA rules for work hours, rest time, etc. In short, the cost and constraints that determine each pairing is an extremely complicated nonlinear, non-smooth, non-convex, nonplussing mixture of on-duty + rest + reliability + crew work life balance + other path-dependent factors. 

Good quality pairings can be created using a dynamic column-generation procedure. A column here represents a pairing, and during an optimization run, thousands of pairings are generated and selected from among the many trillions of possibilities using a "branch and price" scheme. Eventually a subset of the generated columns are selected such that every flight in the time-space network is covered by exactly one pairing, and such that the total cost is minimized (where the actual 'cost' being optimized is a combination of a hundred-odd objective function components).

Note: An airline crew-pairing solver's set partitioning problem is arguably the easiest optimization model in the world to represent on paper:
{SPP: Minimize cx : Ax = 1, x binary, x ∈ X}

In contrast, the real-life computer code used to manage this simple model can easily run in the order of many hundred thousand lines (largely due to the complicated and detailed nature of X), with expensive parallel-processing machines, and an embarrassment of CPLEX licenses to match. Why? A small percentage point reduction in crew pairing cost (cx) can easily recover all these infrastructure costs and then some. Since crews get paid the max of planned and actual costs, these numbers are actual bottom-line savings. Furthermore, such scheduling tools can be mission-critical systems that run 365 days a year, requiring O.R product support staff to be on call like primary care doctors. 

Solution Approach
Downstream systems, and not of immediate interest to this post, combine these pairings into actual monthly schedules with pilot, first officer, and flight attendants assigned by solving yet another set-partitioning problem. Thus cost-effectively scheduling even a single resource requires a sophisticated solution approach. In those days, academic literature often contained interesting approaches that looked good on paper but proved to non-implementable because of the highly context-driven and data-driven nature of this problem that requires developers to pay careful attention to problem size, exploit special structures in SPP and X, and keep in mind certain carrier-specific objectives. On the other hand, interested readers can refer to this journal paper - the first one I read on the job. It is also the #1 cited INFORMS paper based on a recent ranking. Today, this portion of airline planning is typically outsourced to specialized airline crew-scheduling software vendors.

Pairings and Strings
A set of pilot pairings tell you, for any given LEG flight: the in-bound flight that the pilots arrived in, and the out-bound flight on which they will depart. Of course, if a LEG flight schedule is built with lots of tight turns (say 30 minutes or less) at an airport, then it makes sense from the point of view of operational reliability and cost, for the crews to stay with, and depart on the same airplane, assuming of course, that the aircraft routers will assign the same aircraft (tail number) to these flights. This is a resource synchronization rule: Keep crews and aircraft together to the extent possible while also keeping costs low and satisfying all contractual duty, rest, and maintenance rules.

The FA scheduling system can take pilot pairings as inputs to try and keep the FAs with the pilots. Unfortunately, since these crews have different operating hours and safety rules give the varying nature of their job descriptions, very often, pilots and FAs have to break up and depart on different flights. This is an operational nightmare when you are trying to work with tight turns to compete with LCA, and actual costs shoot up if you 'over-synchronize'. To make matters worse, the strings (partial aircraft maintenance routing solutions) also have to be tightly woven with crew pairings. It's a planners nightmare.  Believe it or not, LEG folks actually built much of these schedules separately ("legacy approach"), with each scheduler anticipating what the other person will do, often coordinating via phone-calls in this multi-person game. Result: an expensive and sometimes operationally brittle schedule that increased operational costs and ate up some customer goodwill, which eventually resulted in the plug being pulled. In the next part, we'll see how LEG's OR team analyzed this problem.

to be continued .....

Conflict Resolution - 3: Contextual Optimization

Asimov's Zeroth Law
We continue this discussion from where we left off a few weeks ago: robot-ethics and how Asimov's robots resolved conflicts. A key update to Asimov's original three laws is the inclusion of the zeroth law (from Wikipedia):

"A robot may not harm humanity, or, by inaction, allow humanity to come to harm."

A fundamental difference between this law and the others is its abstract specification, with little clue on how it will be implemented. Furthermore, by giving this law the highest priority, Asenion robots are designed to first and foremost safeguard 'humanity', while also minimizing injury to individual humans and themselves as secondary and tertiary objectives. A robot is given the difficult computational task of proving that the cost of hurting a human is less than humanitarian benefit derived from an alternative action, and must do so within a finite amount of time.

The Universal Conflict-Resolution Model
Immanuel Kant's 'categorical imperative' is an example of an universal conflict resolution model that furiously strives to be context-free, and one that has greatly influenced western thought. Similarly, the Ten-Commandments' "Thou shall not kill" is absolute. Any machine that includes this hard constraint would be unable to kill, even defensively in order to protect a large number of humans under threat. Kantian rules are easy to 'encode-and-forget' within machines and systems since one does not have to ever worry about the context of its application. As we saw in the previous post, the robotic laws are context-free and Kantian in design. The original three laws operated as hard, must-satisfy constraints and necessary conditions. Per Wikipedia, Kant's

"perfect duties are those that are blameworthy if not met, as they are a basic required duty for a human being."

The problem of course is that (see prior post), the rigidity of all-hard rules is not practical and later versions of Asenion robots appear to additionally operate based on the concept of Kant's imperfect duty (again from Wikipedia):

"unlike perfect duties, you do not attract blame should you not complete an imperfect duty but you shall receive praise for it should you complete it, as you have gone beyond the basic duties and taken duty upon yourself"

Thus, imperfect duties are soft, rather than hard constraints, and the aim is to maximally satisfy (minimally violate) requirements. However, the optimization weights that trade off the degree of importance assigned to each of the Asenion robot's 'imperfect duties' are hard-coded, and additional context-specific inputs are required from humans to achieve a satisfactory result in resolving a dilemma. The robots have no authority to perform context-specific conflict resolutions on their own.

How then does this zeroth law practically work in Asimov stories? It's Kantian abstraction is incomprehensible to all except a couple of enlightened robots (with telepathic ability, no less). In general, the zeroth law remains useful only on paper for most of the robots.

Contextual Ethics: The Indian Way
Rajiv Malhotra's path-breaking book "Being Different: Indian Challenge to Western Universalism" provides a fascinating contrast between the traditional Indian approach of 'contextual ethics' (CE) that arose from its Dharmic thought system, and the 'context free ethics' that largely guides the western approach (for an earlier post based on his work, see here. Some of the methods in this post are applications of ideas in this book). From an optimization perspective, we can think of a CE-embedded robot as one that maximally satisfies a combination of hard, soft, and firm constraints, where a 'firm' constraint refer to a hard constraint that is minimally and temporarily relaxed, depending on the specific context of the dilemma, and for the benefit of the 'greatest good', at the expense of incurring a context-specific penalty. This flexibility must not be confused with moral relativism - where a set of soft constraints are tactically and optimally manipulated according to context to maximize some convenient self-serving objective. 'Optimally timing an apology' can be thought of as an example of a non self-serving, contextual optimization model.  It is worth doing a deep dive into this concept, by reviewing some passages in the aforementioned book:

"The frequently leveled charge of moral relativism against this contextual morality is inaccurate, because the conduct and motive are considered consequential in judging the ultimate value of statements.

.... Dharmic ethics are formulated in response to the situation and context of the problem in a way that makes Western ethics seem unduly codified, rigid, monolithic and even simplistic. A.K. Ramanujan, in his influential essay 'Is There an Indian Way of Thinking?', uses the terms 'context-free' and 'context-sensitive' to contrast the West and India in their respective approaches to ethics: "Cultures may be said to have overall tendencies to idealize, and think in terms of, either the context-free or the context-sensitive kind of rules. Actual behavior may be more complex, though the rules they think with are a crucial factor in guiding the behavior. In cultures like India's, the context-sensitive kind of rule is the preferred formulation" ....

.... Dharmic traditions, on the other hand, have long sought to arrive at truth by balancing universal truths and acts with those that can be determined only in the context in which they occur. Dharmic cultures have thus evolved to become comfortable with complexity and nuance, rejecting notions of the absolute and rigid ideals of morality and conduct....

....dharmic thought offers both universal and contextual poles – not just the latter, as that would be tantamount to moral relativism."


The dharmic approach lies in between an "all-soft constraint" and the Kantian "hard and soft constraint" approach to decision optimization.


Applying contextual optimization
Asimov's telepathic robot Giskard formulates and solves a probabilistic optimization problem where it trades off the opportunity cost (in terms of human lives) against the expected benefit to humanity. However the degree of uncertainty in this conflict-resolution model is too high and the robot eventually crashes. This episode comes across as an example of applying the CE approach to resolve a dilemma. The great Indian epics - the Ramayana and the Mahabharata, contain several brilliantly narrated instances of contextual conflict-resolution. Indian sci-fi movie buffs would not be surprised to know that George Lucas' Star Wars was inspired by the Ramayana.

Contextual optimization in the specific area of 'mathematical decision support software' would mean: allowing the rules of engagement to be configurable depending on the context. For regular users, advanced settings are greyed out, with only universal (default) rules enabled. Only super users, who are well-trained and comprehend the nature and consequences of the beast, get to work with 'firm' constraints, and on rare occasions. For example, an airline crew schedule optimization system should be configured to satisfy contractual and FAA rules, except during emergencies (e.g. post 9/11 recovery) where 'crew welfare' is only achievable by overriding one or more of these rules. Practical decision support systems should be carefully designed to allow such controlled contextual optimization.

Amending the Zeroth Law: The Dharmic Robot
The four laws do not quite protect the rest of the cosmos (e.g., from humanity) given their anthropocentric nature. From an Indian point of view, this gap can be closed by modifying the zeroth law based on the contextual ethics of dharma. Rajiv Malhotra, in his book, provides the etymology and a working definition of dharma:

"Dharma has the Sanskrit root dhri, which means 'that which upholds' or 'that without which nothing can stand' or 'that which maintains the stability and harmony of the universe'. Dharma encompasses the natural, innate behaviour of things, duty, law, ethics, virtue, etc. For example, the laws of physics describe current human understanding of the dharma of physical systems. Every entity in the cosmos has its particular dharma – from the electron, which has the dharma to move in a certain manner, to the clouds, galaxies, plants, insects, and of course, man. Dharma has no equivalent in the Western lexicon."

In such a framework, Asimov's laws would delineate a robot's various dharmas. At the highest level, we can require that a robot abide by the following fundamental law, that is based on an ancient Indian text:

"Non-harming is a robot's highest priority, except in the defense of dharma"

Conflict-resolution is always performed by first applying this highest dharmic principle and customizing it to the specific context. Note that by operating on the fundamental dharmic principle of least harm, a robot would usually satisfy Asimov's zeroth law, albeit in a context-specific manner, while also being in harmony with the original laws, as well as any new laws that get written in the future. Interestingly, the Hippocratic oath of medical doctors is based on a similar idea that represents a non-negative bound: "do good, or at least no harm".  If complex systems, new drugs, etc., are designed by always keeping this fundamental principle in context, it may well minimize the risk of catastrophic failure.

Optimally Shoveling Snow

This post attempts to compare the efficiency of various approaches used to clear a rectangular area (L > B) of snow, using a good old manual shovel of width w, always moving in straight lines. We assume that the gradient of the area is zero, and that h inches of snow has fallen uniformly over the field. We also ignore the capacity of the shovel for now. The total snow that must be banked at the borders of the field is constant = LBh. Also total working distance traversed ~ LB/w.

Snow shoveling can be a grueling exercise for some, and public health departments always publish safe shoveling tips during the winter season. This is especially important for people with potential heart problems or back problems. Hopefully analyses like these will help (I'm sure people have looked at this before, but just in case ...).


Our objectives are to minimize:
1. Total work done = Total snow moved X distance moved.
2. Number of Stop-Starts with a fixed unit cost k incurred
3. Total work time in cold weather.

Model:
Given an area bx, and assuming we are moving along x, the snow nearest the starting point must be moved the maximum distance, and this distance linearly reduces to near-zero at the end of x. Thus (introducing an integral sign in this blog, yikes):
a) total work done = ∫wxdx = wx²/2
b) Unit stop-start cost = k

Approach 1: Shoveling east-to-west (along L, banking at the west end) or bi-directional traversing:
total number of passes =  B/w
total work done = (B/w) * wL²/2 + kB/w = BL²/2 + kB/w



Note that the effort level is independent of the shovel width. A wider shovel requires fewer passes but more effort per pass.

Approach 2: Shoveling south-to-north (along B, banking at north end) or traversing in both directions:
total cost = LB²/2 + kL/w

Thus, the second approach reduces the total work done by a factor L/B. However, this reduction is achieved at the expense of more stops (which may not be a bad thing from a health perspective; k may be negative or zero for some). If L = 2B, the second approach requires 50% less effort, but requiring twice as many passes. 

Approach 3: I observed that my Canadian neighbor divides up the rectangle into two areas, always starts from the middle, and creates banks on both sides. If we bank the snow at the south and north ends:
total number of passes =  2L/w
total cost = LB²/4 + 2kL/w

The third approach is twice as effort-efficient as the second approach, at the expense of making twice as many banks, plus an additional penalty of traveling to the mid point after each pass. On the other hand, you are also less likely to run into capacity problems, since the material handled per pass is the lowest. We can also verify that among all starting points chosen for approach-3, the mid-point (B/2) yields the minimal effort. In general, any of these approaches become preferable in terms of total cost, depending on the chosen objective function. For example, we could modify approach-3 and bank snow close to the ends the field (< B/2) at their respective (west or east) ends at the expense of adding more stop-starts.


Time Considerations

If the shape of the field changes, the approaches may change, but it appears that a good principle to follow is to limit the material moved between successive bankings. In particular, we want to identify the shortest Euclidean path between every point in the field to the nearest feasible bank, and purely from an effort perspective, shovel snow along a path to a feasible bank that is optimal for all points on that path. This seems to reduce the working duration taken as well: 
Suppose we maintain constant momentum (i.e. mass X velocity = constant) during a pass, our velocity will inversely proportional to the amount of snow accumulated (e.g. 1/wx). Total time per pass ∝  ∫wxdx = wx²/2, which is proportional to the effort.

For such a velocity model, approach-3 also appears to be optimal from the time perspective. If we assume constant velocity, then the three approaches take the same working time, ignoring the shovel-free travel time required for approach-3. Among these alternative time-optimal approaches, the third approach does the best on the effort-objective.


Lawn Mowing
Non-propelled, walking lawn mowers with bags? Regardless of the traversing method adopted, our effort (per our chosen model) increases as the square of distance until we empty our cut-grass. If we adopt approach-2 or 3, we have more banking opportunities.  However, approach-3 may result in a mowing pattern that won't look so nice, so approach 2 may be more preferable. Operating a self-propelled mower with built-in mulching may reduce our effort considerably.

Update 1: Post "Nemo" Shoveling, February 9, 2013
A truly optimal solution is to have a kind and wonderful neighbor. After three hours of efficient shoveling (h = 24 inches) and 40% completion, Kevin zipped in with his snow plough and finished the remainder in 5 minutes before going off to help others. I now have to find the best way to thank him.

Update 2: (WSJ, via @ThaddeusKTSim), February 10

 (source link: online.wsj.com)

Conflict Resolution - 2: Asimov's Laws of Robotics

Asimov's Laws of Robotics
The introductory discussion on prioritized business rules can be read here. This sequel treats Asimov's sci-fi laws of robotics as a set of prioritized business rules within a decision support system (DSS). We try to learn some 'field lessons' from stories that revolve around violations of one or more of these laws. The three (original) laws postulated in the 1940s and the stories cited in the Wikipedia page are used in this post:

1. A robot may not injure a human being or, through inaction, allow a human being to come to harm.
2. A robot must obey the orders given to it by human beings, except where such orders would conflict with the First Law.
3. A robot must protect its own existence as long as such protection does not conflict with the First or Second Laws.

We can think of these anthropocentric laws as 'Asenion' robotic design constraints in the sci-fi world which, among other things, seek to prevent the creation of Frankenstein-type automatons. Every robotic decision must satisfy these constraints in normal operating conditions. As the laws imply, a lower priority rule can be violated only to prevent the counterfactual infraction of a higher-priority rule. Asimov's robot stories cover situations where a rule is violated for no immediately apparent reason, or a lower-priority rule appears to override a higher priority rule. A robot typically 'crashes/auto-destructs' when faced with an irreconcilable conflict ("damned if you do, damned if you don't").


(Trivia: Robot Chitti in the Indian movie Endhiran is obviously not Asenion. A portion of the plot in the movie bears some resemblance to a story-line in 'Robots of Dawn')

Inadequacy of rules for complex systems

Anticipating 'bugs', and identifying/resolving potential conflicts (low-probability high consequence (LPHC) events in particular) within and between interacting components of a complex, synthesized system can be quite challenging. Recent examples: Boeing 787 Dreamliners, offshore oil rigs, or the western financial system. The positronic robot is no exception. Roger Clarke notes: "Asimov's Laws of Robotics have been a very successful literary device. Perhaps ironically, or perhaps because it was artistically appropriate, the sum of Asimov's stories disprove the contention that he began with: It is not possible to reliably constrain the behavior of robots by devising and applying a set of rules."

(picture link source: Wikipedia)

Hard Constraint Violation

Stories around initial designs of Asenion robots appear to revolve around hard-constraint satisfaction. Humans forget that a telepathic robot also protects humans from mental harm. It often prefers to tell the questioner what she/he wants to hear, violating the second law by not responding with facts, in order to satisfy the higher priority constraint of not injuring human minds 'by dashing hopes'. In the end, it runs into an irreconcilable constraint violation when it can neither speak nor be silent without hurting somebody mentally. It 'dies', taking its telepathic secrets with it.

An expensive decision support system that provides a 'null' response (or '42') after running for a long time can be irritating. There once was a user who loved a DSS when it worked, but would question slow-failures, asking why the application was designed to go through bugs and not around them.

Soft Constraint Violation
A more sophisticated robotic design in another story employs an optimization model. The laws are encoded as soft constraints using potential functions that take into account the level of importance of the order given, the priority of the law, and the degree of constraint 'violation'. Consequently, a 'truant' robot that is faced with a conflict between the second and third laws determines an optimal (equilibrium) solution to the corresponding weighted cumulative violation minimization problem that causes it to continually run around in a circle centered at the origin of the conflict. Doing so prevents the robot from completing a routine task it was assigned. Unfortunately, the robot is uninformed that the delay in the completion of this task is increasingly endangering human lives on the planet. An initial idea of the engineers to break this cycle by perturbing the potentials in the objective function merely shifts the optimum and alters the radius of the Robot's circle, but doesn't help them one bit. Finally, an engineer put his own life in visible jeopardy to inject high-intensity counter-potential (associated with the first law) into the objective function, and is ultimately able to resolve the conflict without loss of life.

This story reminds us of the problems associated with careless softening of 'hard' constraints that attenuates or randomizes the sensitivity of the model's response to priorities and scarce resources ("noisy duals"). The user is left with the unpleasant task of figuring out how to tweak the various 'potential' coefficients for a given problem instance in order to achieve the best results. In practice, an artful mixture of hard and soft constraints usually results in a more manageable and usefully responsive system.

Side-effects of Violation
Users often turn a few knobs in a DSS, look at the results, and wonder "why did this weird thing happen?". This leads to discussions that often unlocks hidden benefits for the customer. In an Asimov story, a robot determines that it is optimal for it to take over the management of delicate instrumentation that is vital to human survival on a planet. It deliberately disobeys humans and gangs up with other robots to permanently banish humans from the control room. The engineers are initially exasperated by this behavior ("is it a bug or a modeling issue?") before a data-driven realization of the life-saving benefit of 'violating' the lower-priority rule calms them. In practice, it is useful to identify unnecessary legacy constraints, if any, that are employed.

Rule violations can also result in undesirable side effects. In the world of airline scheduling, careless rule-changes put in to fix one problem can result in bizarre (hilariously alternative-optimal) schedules for pilots and planes that send them on a wild-goose chase all over the continent. In Asimov stories, engineers discover that a conflict-stressed robot's positronic brain turns it into a practical joker, a prophet, or a wacky friend who plays tag in a life-or-death situation.

Conflict resolution is not just about figuring out what went wrong in the model - a relatively easy task given the excellent solver tools that are available in the market today. It also involves the task of usefully and legibly mapping this change in the model's response back to real world causals. The more interlinked the decision variables and business rules in the system, the more difficult this latter task can become.

In memory of the late Tamil writer Sujatha.

Being Optimally Sorry: When to Apologize?

A Delayed Apology
This post examines modeling ideas related to the timing of an apology in a two-person scenario that results in a maximally effective 'sorry'. We optimize timing here not to maximize own benefits (user optimal), but on the basis of mutual respect, to express regret and maximally repair the damage in a timely manner that most helps the subject (recipient optimal). We start with the findings in Frank Partnoy's book "Wait: the Art and Science of Delay". It's one of the many useful books in the last couple of years that analyze human decision making. We introduce a mental decision support model for a timely apology that is derived from decision analytical methods employed in an industrial setting.

Objectives and Constraints
Justice delayed may be justice denied, but an apology that is optimally delayed may not be such a bad thing. The 'Wait' book recognizes the existence of a suitable time to apologize, and notes that the fastest apology in not necessarily the most effective. Given that we may have to apologize more than once, in general we have to determine an optimal trajectory of timed apologies. Thus, our goals are to:
i)   apologize at least once,
ii)  in a timely manner, and
iii) within a finite time horizon, such that
iv) a measure of the recipient's benefits is maximized

'Wait' notes:
"... Saying you are sorry is always better than not apologizing at all. But as with the first study, the students felt better about a delayed apology: “Improvement in the late apology condition was significantly greater than improvement in the early apology condition.” In fact, a statistically significant improvement in the students’ reactions occurred only in the late apology condition, when there was a chance for them to discuss what had happened and why. Overall, these studies suggest that the relationship between apologies and timing follows a “bell curve” distribution: effectiveness is low at first, then rises, peaks, and ultimately declines."

(It seems that these ideas are related to the complementary 'problem' of delivering the most time-effective 'Thank You')


We can see that the timing-effectiveness curve described in the book extract above is related to the subject's level of distress/angst (which we represent as 'entropy') that follows a similar trajectory of rise, cruise, and a gradual demise. Depending on the person, the 'cruise' and 'demise' portions can last long and result in a very fat-tailed distribution. But before we get into 'when', a quick comment from the book on the what/why/how questions:
".... effective apologies typically contain four parts: 
1. Acknowledge that you did it. 
2. Explain what happened. 
3. Express remorse. 
4. Repair the damage, as much as you can."

Searching for the Optimal Timing
'Wait' notes:
"The art of the apology centers on the management of delay. For most of us, the lesson is that the next time we do something wrong to a close friend or family member, or say something at work we wish we could take back, we should try to imagine how the victim might react to an apology tomorrow instead of today, or in a few hours instead of right now. If delay will give a friend or relative or coworker a chance to react, to voice a response and prepare themselves to hear our regret, the apology will mean more later than right away."

In other words, the timing has to take into account where the subject is located in their entropic life cycle: is the person likely to be getting angrier by the hour now (positive entropic gradient), or has reached the peak and is calming down (negative entropic gradient). To formulate a model based on these observations, we borrow ideas from a classical inventory management problem analyzed in retail operations research: Markdown Optimization (MDO).

An Optimization Model
MDO is employed to manage an inventory of short-life cycle (SLC) products that are manufactured pre-season, with the (sunk) costs paid up-front. Thus MDO typically focuses on total revenue earned in-season. Analogously, we already messed up in the beginning incurring an irreversible cost, and thereafter it costs relatively little to issue a sincere apology.  Retailers employ a cadence of optimally delayed price cuts to smartly boost the end-of-season demand rate so as to maximize revenue over the remaining life of the product. Like MDO, we eventually have to solve an entropic inventory depletion problem: optimally alter the entropic gradient via one or more carefully timed apologies, which will (ideally) reduce the inventory level to zero within a finite time period.

Disclaimer: The postulated model is not assumed to be the most suitable or even a "correct" one for this problem, but merely a useful starting point. Some brief comments on the modeling elements, next.

a. Life-cycle of the entropy
SLC products (like designer fashion apparel) often have little to no historic data early in the season, and retailers may borrow results for a comparable historical "like-item" to produce an initial prediction and then continually update their sales projection based on in-season demand. Here, we play the role of a 'like-item' and place ourselves in the recipient's shoes to better appreciate the degree of distress caused and the impact it will have on the recipient over time. The entropy level is an uncertain quantity that must be learned, but its 'mean value' is assumed to representable using an approximately concave function like the one shown in the figure below. Note that unlike the MDO case where inventory is always non-increasing, entropic inventory initially increases before gradually decreasing.

b. Elasticity of the entropy with respect to an apology

Elasticity ~ % change in entropy / % change in regret and effort, as perceived by the subject
 
A simple model like the inverse square law that abounds in nature (elasticity = -2) may be a good starting point. Ill-timed and empty-sounding apologies may have zero elasticity and do little to reduce entropic inventory. A careless apology can result in an entropic spike ("adding insult to injury"). On the other hand, an apology that is 'deep and sincere' and well-timed can be expected to have a calming effect.

c. Timings
Optimally timing a single apology requires impeccable timing. On the other hand, randomly distributed, and incessant apologies may not be helpful either. A premature apology (e.g. around an increasing entropic gradient) that kicks the can down the road is a greedy approach that may be counter-productive. Thus, optimally timing multiple apologies can require a degree of coordination between decisions. Today's apologize-or-delay decision will impact the timings of future decisions, so we have to holistically manage the impact on the entropic life-cycle. 

Often, despite our best efforts, the damage can never be fully repaired. Note that our objective function was setup to be indifferent to personal benefits. To paraphrase a profound Indian saying: "You have the right to optimize, but not to the fruits of your actions". Regardless of the outcome, a sincere and optimally timed apology is good Karma.

Business Rule Conflict Resolution: Needles in a Haystack

This post uses the retail industry as a sketchpad to illustrate some simple but useful conflict resolution ideas employed within optimization methods in practice.

Least Infeasible Answers
Retailers have to address a myriad of business rules while optimizing merchandising, pricing, shelving, replenishing, and assortment planning decisions. Operations Researchers build mathematical models that translate these rules into a combination of soft constraints (goals, objectives, Key Performance Indicators (KPI)) that have to be maximally satisfied, and hard constraints that must be absolutely satisfied.
Conflicting hard rules can prevent the optimization application from returning a recommendation. The retailer would like to know which among their long list of rules is/are the root cause of this conflict so that they can address them appropriately and re-evaluate. Commercial solvers like CPLEX (and GUROBI, among others) can perform an infeasibility analysis to provide useful information pertaining to IIS (Irreducibly Inconsistent Sets), and based on optimizing a user-parameterized penalty function, can also find a minimally infeasible answer. These are non-trivial problems. (Try explaining to a senior pilot why a more junior pilot was assigned that plum schedule filled with all those long Hawaii layovers!) JW Chinneck has an entire book dealing with such issues.

User Experience
The liberal-arts degree armed retail store manager (RSM) has a business to operate and sales targets to achieve. Many a RSM care little for iterative conflict refinement and resolution and have even lesser patience to set up paramterized penalty functions. He/She often desires that the analytics software product automatically flag problematic rules and always return the "best possible" answer(s) and be done with it. 'Null' or 'Try, try again' is not an option. A positive user experience combined with the ability to "help me earn a bonus or you are toast" is correlated with continued patronage. In such situations, the decision analytics plumbing must be designed in such that it can, with little to no external guidance, produce practically useful diagnostics that pinpoints infeasibility and generate 'best fit' solutions that are readily business-comprehensible to the average user. After all, the business application interface and functionality is not primarily meant to be a analytics-tutorial warp drive thingy.

 (picture link source: anandtech.com)

Actions Express Priorities - Mohandas Gandhi
A store manager may have a high-level goal in mind such as: find a minimally business disruptive resolution of conflicting business rules. Toward this, several retailers employ prioritized business rules that result in a special haystack structure that invariably makes the conflict identification and resolution process more transparent and easy to manage. What would this decision problem structure look like? Borrowing computer-ware terminology (for example), it would be akin to an ordered collection of hard constraints, soft constraints, and firm constraints. Patented methods that gainfully exploit related ideas have been used to build retail optimization products that turned out to be successful in the global marketplace. On the other hand, the efficacy of such resolution methods can be context-specific and is not necessarily designed to yield universally scalable solutions. If your job is to solve only your company's or industry's specific problems and not that of all current and future corporations in the galaxy and beyond, such methods can be handy.

Priceless Pricing

Listening to Jazz feels like time well spent, despite a limited understanding of this beautiful American art-form. John Coltrane and Miles Davis are especially close to heart given their affection for India's native music that also influenced how they played. Miles is recognized for paying attention to the 'space between the notes' as noted in this blog:


"Among Miles’ trademarks was his emphasis on the space between the notes as much as the notes themselves.  Silence was key to his music.  He was so cool he didn’t feel the need to jam up every measure with noise.  Sometimes it’s better to just stand there and not do something."

It is well known that silence is an important tool in the hands of an expert stand-up comedian or a novelist. In the world of ORMS, optimally scheduling working hours for airline crews is really about paying close attention to their silent periods of rest. In the world of pricing optimization, it is imperative to recognize that a customer always values something else in the product in addition to price. To usefully optimize price, we must accurately identify its priceless attributes.

Update: January 11, 2013
Richard Marcus, former CEO of Neiman Marcus:  “Price is important, but quality is remembered long after price is forgotten, as are qualities of uniqueness and originality.”

Update: February 06, 2013
Competing beyond price to maximize customer loyalty: "...“Attractive prices are an effective means to get people in the door, but it’s not enough to maintain loyalty, drive future purchases, and generate customer recommendations..."

Update: May 30, 2013
Price Perception: " ....it’s not about price, it’s about what am I getting for that price"