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 .....