Decision aid methodologies in transportation Lecture 6: Miscellaneous Topics Prem Kumar prem.viswanathan@epfl.ch Transport and Mobilit Laborator
Summar We learnt about the different scheduling models We also learnt about demand-suppl interactions in the form of revenue management concepts Toda, we will see further application of revenue management to airline industr Some more examples of integer programming formulations Lastl, some new applications
Revenue Management: H&S Airline CDG CHF150 Path Leg 1, CHF 100 GVA Leg 2, CHF 100 ZRH Given A passenger intends to book a seat on CDG-GVA Question Should ou sell it or should ou wait to sell the ticket for a passenger intending to book CDG-ZRH for a higher revenue? Complexit Millions of itinerar
Airline Revenue Management Leg Optimization - Set explicit allocation levels for accepting bookings on each flight leg Network Optimization - Determine the optimal mix of path-class demand on the airline network
Airline RM: Network Optimization Model LP model to maximize revenue subect to capacit and demand constraints Network consists of all legs departing on a given departure date (a few thousands) and an path-class with a constituent leg departing on this date (up to a million) Model considers the following to determine demand: cancellation forecast no show forecast upgrade potential The displacement cost of a leg/cabin is the shadow price of the corresponding capacit constraint of the LP
Airline RM: Network Optimization Formulation n Path-Classes: f 1, f 2,..., f n fares d 1, d 2,..., d n demand x 1, x 2,..., x n decision variables m Legs: c 1, c 2,..., c m capacities Incidence Matrix A=[a i ] mxn a i = 1 if leg belongs to path i, 0 otherwise LP Model: Maximize Σ f i x i Subect To Σ a i x i < c = 1,2,..., m capacit constraints 0 < x i < d i i = 1,2,..., n demand constraints
Integer Programming: More Formulations Consider the following mathematical formulation: min c T x Ax b x 0 View this formulation as the one where x indicate different options and c T the corresponding costs. However if an option is selected, a fixed cost is incurred b default PROBLEM: x = 0 or x k How to formulate this?
Integer Programming: More Formulations Use a binar auxiliar variable = 0, for x 1, for x = 0 k Add the following constraints: x M (M is an upperbound on x) x k {0,1}
Integer Programming: More Formulations This can be applied even when x is not necessaril an integer minimize C(x) where : Ax = b x 0 C(x i ) = k i + 0 c i x i for x for x i i = > 0, 0. Use auxiliar variable i = 0, for x 1, for x i i = > 0 0 Add these constraints x C( x i i M i ) = k {0,1} i i i + c i x i
Integer Programming: More Formulations Consider the following constraint: x + x 2 1 If the constraint has to be absolutel satisfied, it is called a hard constraint However in some situations, ou ma be able to violate a constraint b incurring a penalt Such constraints are called soft constraints and the can be modeled as: T minimize c x + x1 + x 2 5 + Y x 0, Y 0 Y 100 5
How to consider variables with absolute values? Consider this: How to solve this tpe of formulation? Integer Programming: More Formulations free 0, x b x a min t,t t t,t t + = + + + = = t t t t t t 0 0, 0, ) ( min,, + = + + + + t t t t t t t t t t x b x a
How to treat disunctive programming? A mathematical formulation where we satisf onl one (or few) of two (or more) constraints Integer Programming: More Formulations k k k p x x or p x x w x min {0,1} ) (1 M p x x M p x x w x min 2 k 1 k k
Integer Programming: More Formulations We started describing MIP with Transportation Problem But the problem can be solved with SIMPLEX method. Yes! Consider a mathematical formulation min c T x Ax b x 0 Suppose all coefficients are integers and constraint matrix A has the propert of TUM (Total UniModularit) TUM implies that ever square sub-matrix has determinant value as 0, -1 or 1 There exists an optimal integer solution x* which can be found using the simplex method
Optimization at Airports
Airport Gate Assignment: Obectives Given a set of flight arrivals and departures at a maor hub airport, what is the *best* assignment of these incoming flights to airport gates so that all flights are gated? Gating constraints such as adacent gate, LIFO gates, gate rest time, towing, push back time and PS gates are applicable
Airport Gate Assignment: Problem Instance One of the largest in the world Over 1200 flights dail Over 25 different fleet tpes handled 60 gates and several landing bas Around 50,000 connecting passengers
Terminolog Adacent Gates: Two phsicall adacent gates such that when one gate has a wide bodied aircraft parked on it, the other gate cannot accommodate another wide bod Gate #1 Gate #2
Terminolog Market: An origin-destination pair Turns: A pair of incoming and outgoing flights with the same aircraft or equipment Gate Rest: Idle time between a flight departure and next flight arrival to the gate. Longer gate rest helps pad an minor schedule delas, though at the cost of schedule feasibilit PS Gates: Premium Service gates are a set of gates that get assigned to premium markets tpicall where VIPs travel
Mathematical Model Parameters a i : scheduled arrival time of turn b i : scheduled departure time of turn (k,l): two gates restricted in the adacent pair E k E 1 1 1, : sets of equipment tpes such that when an aircraft of a tpe in E l k is 1 occuping k, no aircraft of an tpe in ma use l; and vice versa. Decision variables x ik {0,1}: 1 if turn i is assigned to gate k; 0 otherwise i {0,1}: 1 if turn i is not assigned to an gate; 0 otherwise E l
Mathematical Model Maximize i T k K C ik x ik C i T i subect to: k K x ik + i =1 i T x ik + x k 1 i, T; k K : a < b + α, a < b + α, i i i x x 1 i, TURNS; k, l GATES; ( k, l) ADJACENT : ai < a ai < b ik + l a < b 1 1 i, i ; ei Ek; e El
Output: Gantt Charts Assigned Flight
Additional Obectives Maximize Connection Revenues This gating obective identifies connections at risk for a hub station and gates the turns involved such that connection revenue is maximized Maximize Schedule Robustness Flights must be gated based on the past pattern of flight delas to provide adequate gate rest between a departing flight and the next arriving flight Maximize Manpower Productivit While gating the flights, emploees could potentiall waste a lot of time travelling between gates
Revenue Management We have a lot of demand for a flight to ORD 10 das from now. A320 tail that is scheduled for it is not big enough Maintenance That A320 tail has to go to ORD for maintenance! With this new rotation m pilot has to switch aircraft, but the connection is too short. Do ou know how much it will cost to use a reserve?!! Crew Ok, I ll swap it with a A321 tail Well, I can change rotations and this tail will end up in ORD after all Ops I ve worked hard esterda and b the end of the da the schedule for the next 3 weeks was perfect Planning Fine! I retime the flight b 10 minutes, so our pilot has enough time I give up Airport Retime the flight?!!! I don t have a gate for this flight if it is 10 minutes later Don t worr about it. ORD is on ground dela for another 4 hours. B the time we sort things out ou ll have to readust the whole next week anwa Courtes: Serge Shebalov, Sabre Technologies
Optimization in Railwas
Applications in Railwas Locomotive Assignment Locomotive Refueling Revenue Management Locomotive Maintenance Platform Assignment Train-design Block-to-Train Assignment
Locomotive Assignment Basic Inputs Train Schedule over a period of planning horizon A set of locomotives, their current locations and properties Output Assignment schedule of locomotives to trains Constraints Locomotive maintenance Tonnage and HP requirement of train Several other constraints Obective Cost minimization
Locomotive Assignment: Some Features A train is tpicall assigned a group of multiple locomotives called a consist that usuall travels together Each train has a different HP and Tonnage requirement that depends on the number of cars attached Locomotives can either pull trains activel or deadhead on them. Locomotives can also light travel. Trains need not have the same dail schedule.
Locomotive Assignment: Mathematical Model Decision Variables Locomotive-Train assignment schedule Active locomotives Deadhead locomotives Light travel locomotives Parameters Locomotive availabilit, maintenance schedule and features Train schedule / time-table and train features Infrastructure features for sections and ards
Locomotive Assignment: Hard Constraints Horsepower requirements Tonnage requirements Fleet size limitations Consistenc of the assignments Locomotive availabilit at ards and sections Repeatabilit of the solution Solution robustness and recoverabilit
Locomotive Assignment: Solution Methodolog Input Data Shorter period Scheduling Problem Complete time horizon Scheduling Problem Solution Two-stage optimization allows us to reduce the problem size substantiall while giving an opportunit to maintain consistenc
Locomotive Assignment: Solution Methodolog Input Data Form traintrain connections that can be served b the same locomotive Determine locomotives for light travel and deadheading depending on locomotive imbalances Determine minimal cost assignment of locomotives Solution Determine the three sets of decision variables using a sequential process.
Railroad Blocking Problem Problem: Origin-Destination of shipments given Each shipment contains different number of cars Train routes and time table known Capacit of the network and trains known Magnitude: Thousands of trains per month 50,000 100,000 shipments with an average of 10 cars (Ahua et al) Design the network on which commodities flow
Comparison with Airline Schedule Design Hongkong Delhi Frankfurt Chicago Zürich Prague Paris Geneva Lon Lisbon Munich Milan Vienna
Railroad Blocking Problem Blocking Arcs Origins Yards Destinations Reference: Ahua et al: Railroad Blocking Problems
Railroad Blocking Problem: Model Decision Variables: Blocking arcs to a ard with origin (or destination) selected, or not Route followed b the shipments along the blocking arcs Constraints: Number of blocking arcs at each node Volume of cars passing through each node Capacit of the network and train schedule Obective Function: Minimize the number of intermediate handling and the sum of distance travelled (different obectives can be weighted)
Railroad Blocking Problem: Problem Scale Network size: 1,000 origins 2,000 destinations 300 ards Number of network design variables: 1,000x300 + 300x300 + 300x2,000 1 million Number of flow variables: 50,000 commodities flowing over 1 million potential arcs Reference: Ahua et al: Railroad Blocking Problems
Railroad Blocking Problem: Complexit Network design problems are complex for man reasons. Apart from the large number of variables, there can be several competing solutions with the same value of the obective function Problems with onl a few hundred network design variables can be solved to optimalit Railroads want a near-optimal and implementable solution within a few hours of computational time. Reference: Ahua et al: Railroad Blocking Problems
Railroad Blocking Problem: Solution Approach Integer Programming Based Methods Slow and impractical for large scale instances Network Optimization Methods Start with a feasible solutions Graduall improve the solution one node at a time Reference: Ahua et al: Railroad Blocking Problems
Railroad Blocking Problem: Solution Approach Start with a feasible solution of the blocking problem Optimize the blocking solution at onl one node (leaving the solution at other nodes unchanged) and reroute shipments Repeat as long as there are improvements. Reference: Ahua et al: Railroad Blocking Problems
Railroad Blocking Problem: Solution Approach Out of about 3,000 arcs emanating from a node, select 50 arcs and redirect up to 50,000 shipments to minimize the cost of flow. Problem instance could be solved for one node using CPLEX in one hour. Reference: Ahua et al: Railroad Blocking Problems
Railroad Blocking Problem: Future This is one of the ongoing research open problems that is currentl being tackled b the railroad industr Of course there are man such interesting problems in railwas and we could give example of onl two in this lecture