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....