Choice-based flexible product pricing model · Choice-based flexible product pricing model D5.2...

Choice-based flexible product pricing model

D5.2

COCTA Grant: 699326 Call: H2020-SESAR-2015-1 Topic: Sesar-05-2015 Consortium coordinator: Univerzitet u Beogradu – Saobracajni fakultet Edition date: 30 January 2018 Edition: 01.00.00

EXPLORATORY RESEARCH

EDITION [01.00.00]

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The opinions expressed herein reflect the author’s view only. Under no circumstances shall the SESAR Joint Undertaking be responsible for any use that may be made of the information contained herein.

Authoring & Approval

Authors of the document

Name/Beneficiary Position/Title Date

Arne Strauss/UW Principal researcher (OR) 30/01/18

Stefano Starita/UW Post-doc researcher (OR) 30/01/18

Radosav Jovanović/UB-FTTE Principal Researcher (ATM) 30/01/18

Nikola Ivanov/UB-FTTE Researcher (ATM) 30/01/18

Goran Pavlović/UB-FTTE Researcher (ATM) 30/01/18

Frank Fichert (FF)/HW Principal Researcher (Economics) 30/01/18

Reviewers internal to the project


Obrad Babić Senior Expert (ATM) 30/01/18

Approved for submission to the SJU By — Representatives of beneficiaries involved in the project


Radosav Jovanovic/UB-FTTE Principal Researcher (ATM) 31/01/18

Arne Strauss/UW Principal Researcher (OR) 31/01/18

Frank Fichert/HW Principal Researcher (Economics) 31/01/18

Rejected By - Representatives of beneficiaries involved in the project


Document History

Edition Date Status Author Justification

00.00.01 15/08/17 Initial Draft AS, SS, RJ, NI, GP, FF Initial draft

00.00.02 05/09/17 Draft AS, SS, RJ, NI, GP, FF Draft

00.00.03 28/09/17 Final Draft AS, SS, RJ, NI, GP, FF Final draft, following up the progress meeting

00.01.00 05/10/17 Final AS, SS, RJ, NI, GP, FF Final

00.02.00 10/11/17 Final AS, SS, RJ, NI, GP, FF Answers to the SJU comments on ver. 00.01.00

DELIVERABLE 5.2 - CHOICE-BASED FLEXIBLE PRODUCT PRICING MODEL

© – 2017 – COCTA consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions.

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00.03.00 23/11/17 Final AS, SS, RJ, NI, GP, FF Answers to the SJU comments on ver. 00.02.00

00.04.00 30/01/18 Final AS, SS, RJ, NI, GP, FF Answers to the SJU (additional) comments

01.00.00 30/01/18 Final version AS, SS, RJ, NI, GP, FF Approved and version updated to 01.00.00

EDITION [01.00.00]

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COCTA COORDINATED CAPACITY ORDERING AND TRAJECTORY PRICING FOR BETTER-PERFORMING ATM

This deliverable is part of a project that has received funding from the SESAR Joint Undertaking under grant agreement No 699326 under European Union’s Horizon 2020 research and innovation programme.

Abstract

This deliverable deals with the computational challenges of incorporating trajectory pricing decisions and airlines choice behaviour into the mathematical model.

A parallel is drawn from COCTA to the generic revenue management context to provide a model formulation easier to understand and test. We propose a solution approach based on the idea of re-solving a deterministic approximation of the model several times during the booking horizon. The precision and scalability of the approach is tested with examples of increasingly size.



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Table of Contents

Abstract ................................................................................................................................... 4

1 Introduction ............................................................................................................... 6

2 Mathematical Model ................................................................................................. 7

2.2 Dynamic program’s state ............................................................................................. 11

3 Solution Approach.................................................................................................... 16

3.1 Small-scale example .................................................................................................... 17

3.2 Large-scale networks ................................................................................................... 23

4 COCTA Implications .................................................................................................. 26

5 Conclusions and next steps ....................................................................................... 27

6 References ............................................................................................................... 28

EDITION [01.00.00]

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1 Introduction

In the previous deliverables and publications, we have highlighted how a deterministic model can be used to analyse the effects of capacity decisions on a short (Starita et al., 2016) and longer-term basis (COCTA consortium, 2017). In those works, we operated under the assumption that Aircraft Operators (AOs) select the cheapest/shortest route available and although we included charges in the formulation, no trajectory pricing decisions were explicitly investigated. In this deliverable, we introduce a more realistic formulation, closer to the COCTA mechanism. The concepts of Purchased Specific Trajectories (PST) and Flexible Assigned Trajectories (FAT) are brought in along with a choice model to estimate the likelihood of AOs to choose one product over the other. Furthermore, pricing becomes an active tool which the Network Manager (NM) uses to influence demand so that it better fits with the capacity availability.

The main objective of this deliverable is, therefore, to introduce a comprehensive formulation capable to support the NM’s decision making when it comes to capacity management (overall capacity budget and sector configuration) and product pricing. Such a formulation is significantly complex and requires an ad-hoc solution approach to obtain approximate results.

Within the Operations Research context, the usual way to analyse a solution approach for an optimisation problem is to test it with different datasets. The approach scalability is assessed by evaluating its performances while changing datasets parameters (size, for instance). Some of these datasets are publicly available and used as benchmarks by researchers. Sometimes, the problems analysed bring in new concepts and therefore cannot be tested against the usual benchmarks. In these cases, new datasets have to be built. This applies to the COCTA problem as well, which introduces a set of issues that cannot be tackled by any of the publicly available datasets. Furthermore, it is not practical to develop an Air Traffic Management case study big enough to test the scalability of the solution approach in a short time. In fact, it would require downloading, filtering and editing data of thousands of flights, trajectories and hundreds of sector configurations. This is beyond the scope of D5.2 deliverable. Therefore, we study the problem as a generic revenue management problem. Throughout the deliverable, we highlight several similarities between these two problems, as well as a few differences. Building a case study for the revenue management problem is much simpler and can provide computational feedback that applies to the COCTA problem as well.

The remainder of the deliverable is structured as follows. In Section 2, we present a generic revenue management model which we gradually make more complex by incorporating elements specific to COCTA. In Section 3, we present a small-scale example, propose a solution to solve it and test the scalability thereof. We discuss implications for the COCTA mechanism in Section 4, with conclusions and further steps detailed in Section 5.



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2 Mathematical Model

The mechanism introduced by COCTA can be modelled as a revenue management (RM) problem where only flexible products are being offered. PST and FAT are both flexible products with the first having more stringent margins than the second. To improve readability of this deliverable, we will start by introducing a generic formulation drawn from RM literature (Petrick et al.2012). This formulation will be gradually modified to include the specific details of COCTA mechanism.

As a first step, we consider a problem where a firm dynamically sets the price of its flexible products so as to maximise its expected revenue. Products are sold over a booking horizon split into 𝑇 discrete time periods. The firm has a limited amount of resources 𝐻 = {1,… , ℎ} used to construct the products on offer. The resource budget is represented by vector 𝐵 = {𝐵1, … , 𝐵ℎ}. Arriving customers are divided into 𝐿 segments, indexed by 𝑙. We use 𝐼𝑙 to define the set of flexible products offered to a customer belonging to segment 𝑙. These segments are assumed to be non-overlapping (i.e., 𝐼𝑙 ∩ 𝐼𝑙′ = ∅, ∀𝑙 ≠𝑙′). Each flexible product 𝑖 ∈ 𝐼𝑙 is defined by a set 𝑅𝑖 of specific products 𝑗. A specific product 𝑗 consumes 𝑎𝑗𝑚 units of resource 𝑚 ∈ 𝐻. In a generic problem, products might consume more than one

unit of each resource. Here, we limit us to at most one unit per product in line with COCTA flight/sector unit consumption. Therefore, 𝑎𝑗𝑚 ≤ 1, ∀𝑗,𝑚.

Decision variables 𝑔𝑖𝑡 are used to identify prices charged for 𝑖 at time 𝑡. At the end of the booking horizon, the firm makes the final assignments from flexible to specific products, mindful of the resource budget. These assignments are tracked using binary variables 𝑦𝑖𝑗. In every time period, the

probability that a customer from segment 𝑙 appears is 𝜆𝑙. Given prices 𝒈𝒕, a customer will purchase product 𝑖 with probability given by choice model 𝑃𝑖(𝒈𝒕).

A common approach in RM to model such an optimal control problem is dynamic programming (DP). Using this mathematical solution method, we provide below one possible formulation of the Flexible Products Pricing (FPP) problem based on the following notation:

Sets:

𝐻 Set of resources

𝐼𝑙 Set of flexible products for segment 𝑙

𝑅𝑖 Set of specific products available for flexible product 𝑖

Indices:

𝑙 Segment index

EDITION [01.00.00]

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𝑖 Flexible product index

𝑗 Specific product index

𝑚 Resource index

𝑡 Time index

Parameters:

𝐿 Number of segments

𝜆𝑙 Customer rate of arrival for segment 𝑙

B Resource budget vector

𝑑𝑗 Cost of building product 𝑗

𝑎𝑗𝑚 Number of units of resource 𝑚 used to build specific product 𝑗

Variables:

𝑔𝑖𝑡 Price of product 𝑖 at time 𝑡

𝑦𝑖𝑗 = 1 if flexible product 𝑖 is assigned to specific product 𝑗; 0 otherwise.

𝑉𝑡(𝒀) = 𝐦𝐚𝐱𝒈

∑𝝀𝒍∑𝑃𝑖(𝒈𝒕)

𝒊∈𝐼𝑙

𝑳

𝒍=𝟏

(𝑔𝑖𝑡 − ∆𝑖𝑉𝑡+1(𝒀)) + 𝑉𝑡+1(𝒀), for 𝑡 = 1,… , 𝑇 (1)

𝐦𝐚𝐱𝒚

{

−∑∑𝑑𝑗𝒋∈𝑹𝒊

𝑦𝑖𝑗𝒊∈𝑰 |

|

∑ 𝑦𝑖𝑗𝒋∈𝑹𝒊

= 𝑌𝑖 , ∀𝑖;

∑∑ 𝑎𝑗𝑚𝒋∈𝑹𝒊𝒊∈𝑰

𝑦𝑖𝑗 ≤ 𝐵𝑚, ∀𝑚;

𝑦𝑖𝑗 ≥ 0 and integer

}

𝑡 = 𝑇 + 1 (2)

Function 𝑉𝑡(𝒀) is typically called value function and represents the expected revenue-to-go at time 𝑡, for a given state 𝒀. The state of the problems is a vector with as many components as there are flexible products. The component 𝑌𝑖 represents the number of flexible product bookings that have been received so far. The expression ∆𝑖𝑉𝑡+1(𝒀) is defined by ∆𝑖𝑉𝑡+1(𝒀) = 𝑉𝑡+1(𝒀) − 𝑉𝑡+1(𝒀 + 𝟏𝑖), where 𝟏𝑖 is a unit vector with the 1 in the ith position.

During the intervals 𝑡 = 1,… , 𝑇, flexible products are sold to arriving customers and the firm sets the prices to maximise the value function (1). Every time period is assumed to be sufficiently small such that the probability of more than one request arriving within the same time period is negligible. The end of the booking horizon is denoted by 𝑇 + 1; no more bookings can be made, and the firm decides on the optimal flexible-to-specific assignments (2). We assume 𝑑𝑗 to be the variable cost of providing

product 𝑗. Product assignments are subject to resource capacity constraints.



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The remainder of this section is devoted to gradually move from the generic context represented by the model above to a COCTA-specific formulation.

2.1.1 Customers

In our FPP formulation, we model demand by simply assuming that in every period there is a probability (defined by 𝜆𝑙𝑡) that a customer will appear. In COCTA, customers correspond to flights that need to purchase a trajectory. The set of scheduled flights is denoted by 𝐹 and indexed by 𝑓. Of these flights, the NM knows most of the basic details (Origin-Destination 𝑜𝑑𝑓, scheduled departure time 𝑡𝑜𝑓),

therefore the only uncertainty lays in when a flight appears in the booking horizon and what trajectory product it purchases.

A parallel can be drawn between customer segments and flights. In fact, having the departing time embedded within the product has the consequence of moving from the case where customers make their decisions picking from a common pool of similar products to the one where each customer has its own pool of products to choose from. Therefore, a flight 𝑓 can be interpreted as a distinct segment with its own flexible and specific products. The probability of flight 𝑓 to book at time 𝑡 is given by 𝜆𝑓𝑡.

We assume that scheduled flight trajectory requests arrive with certainty, i.e. ∑ 𝜆𝑓𝑡𝑡 = 1 for all flights

f. As before, time periods are designed sufficiently small such that the probability of an arrival in period t, as expressed by ∑ 𝜆𝑓𝑡𝑓 , is negligible. More details on the products’ choice are given in the next

paragraph. We do not consider booking cancellations as they are rare events which will not likely have a significant impact on the whole mechanism.

2.1.2 Product resources

In the generic FPP model we assume that a specific product consumes a known number of units of each resource. Products and resources in COCTA are more complex. A specific product is a defined trajectory, identified by origin-destination airports, time of departure, sequence of elementary/collapsed sectors and their respective travel times. The notation regarding routes, sectors and airspaces is the same as in the deliverable D5.1 (COCTA consortium, 2017).

Both FAT and PST are flexible products defined by spatial and temporal displacement margins. We

assume that trajectories for each of the products are given as input. With 𝑅𝑜𝑑𝑃𝑆𝑇 and 𝑅𝑜𝑑

𝐹𝐴𝑇 we denote the set of trajectories connecting 𝑜𝑑 within the margin of PST and FAT, respectively. The probability of

a flight to choose a PST is given by 𝑃𝑓𝑃𝑆𝑇(PST and FAT charges). The purchase probability depends on

the relative prices of PST and FAT, amongst other (implicit) factors reflected by a so-called preference value for each PST and FAT. We do not consider the no-purchase option, therefore the probability of

purchasing a FAT is given by 𝑃𝑓𝐹𝐴𝑇 = 1 − 𝑃𝑓

𝑃𝑆𝑇 .

The concept of resource in COCTA translates to the Air Traffic Controls (ATCs’) capacity limitations, measured by means of sector entry count. In other words, a trajectory using an elementary sector at a given time, consumes one ATC capacity unit.

EDITION [01.00.00]

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2.1.3 Capacity budget

In COCTA there is a further layer which differentiates the problem from FPP formulation. Decisions on airspace capacity configurations has a major impact on the capacity limitations of resources. Therefore, we included this in our modelling as described in D5.1. We assumed that the Network Manager can change airspace configurations every 30 minutes, thus directly affecting the maximum number of flights using every portion of the airspaces. For a more detailed explanation of the capacity management and its notation the reader can refer to the deliverable D5.1 (COCTA consortium, 2017)

2.1.4 Objectives: profit and costs

Another key difference between FPP and COCTA is that the Network Manager is not seeking to make any profit. The assumption is that NM is revenue neutral and therefore its aim is to minimise airlines’ costs while recovering capacity provision costs. Consequently, the model will be a minimization problem as opposed to FPP. Airlines costs are computed, as in the D5.1, as the sum of displacement and route charges, i.e. cost of capacity provision. Furthermore, capacity provision costs are estimated using the same rationale as in previous deliverables, i.e. each airspace contributes with a fixed cost and a configuration-dependent variable cost (see the D5.1 for details).

2.1.5 Charging

The generic model does not put any constraints on the charging decisions, hence charges can fluctuate over the booking horizon. Within the highly regulated ATM context this freedom cannot be ensured as it would raise concern about equity. Furthermore, in COCTA the airport-pair charging policy is implemented, de-coupling, in a base case, any pricing decisions from different routes between the two airports (since base charge is the same). Consequently, we assume that the NM chooses PST prices that are non-decreasing in time. The NM also decides on the amount of discount for FAT products. As opposed to PST charges, a FAT discount can increase and decrease without constraints. For a given

period 𝑡, the charge for a PST connecting pair 𝑜𝑑 is given by 𝑔𝑜𝑑𝑡𝑃𝑆𝑇. The price of the corresponding FAT

is 𝑔𝑜𝑑𝑡𝐹𝐴𝑇. We assume that FATs are always cheaper than PSTs. An aircraft type dependent factor 𝑣𝑓 is

multiplied to the charge to reflect take-off weight.

2.1.6 Uncertain demand

In addition to demand known six months in advance, we assume, based on historical data (EUROCONTROL PRC, 2017), that a share (~20%) of the total flights in the model are not scheduled. This set comprises primarily charter flights, Business Aviation flights, portion of all-cargo flights, as well as military and other non-scheduled. Information on these flights become known before the day of operation and therefore those can be modelled like schedule flights. The difference is that for uncertain flights there is a probability of arrival ∑ 𝜆𝑓𝑡 < 1𝑡 , that is, AOs reveal their intention to fly

between two airports at specific time before the day of operations. The rest of uncertain flights (mainly Business Aviation and transatlantic flights) will appear on the day of operation, when the NM might have limited flexibility, depending on the overall traffic and sector opening scheme. The set of these

flights is defined as 𝐹𝐿. We assume that the NM can estimate the likelihood (𝑝𝑜𝑑𝐿 ) that an uncertain

flight will connect airport pair 𝑜𝑑. We refer to the subset of pairs interested by this traffic as 𝑂𝐿. In the formulation we assume type of aircraft and time of departure to be always known. This assumption



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can be easily relaxed by replacing these parameters with their estimated probability distributions. Furthermore, given the late appearance on the booking horizon of these flights, they are charged at

an increased rate 𝑔𝑜𝑑 𝑇+1𝑃𝑆𝑇 . Finally, in this deliverable we assume that these flights must purchase the

FAT product to ensure that we have sufficient flexibility to accommodate those flights, despite the late appearance in the system. This assumption could be subject to changes in the next deliverable (D5.3).

2.2 Dynamic program’s state

In the generic FPP problem, the dynamic program’s state is characterized by vectors 𝒀𝑷𝑺𝑻and 𝒀𝑭𝑨𝑻 of size 𝐹, to keep track of whether a flight has purchased a PST or a FAT product. More specifically,

𝑌𝑓𝑃𝑆𝑇 = 1 if 𝑓 purchases a PST or 𝑌𝑓

𝐹𝐴𝑇 = 1 if 𝑓 purchases a PST instead. Both vectors are initialised to

0 and are updated after a booking. The model stores the revenue collected in scalar 𝑃. It is also necessary to keep track of the PST charges set in the previous time period so as to enforce the non-decreasing assumption.

The overall notation is summarised below:

Sets:

𝑂 Set of origin-destination pairs

𝐹 The set of all scheduled flights, excluding flights which book on the day of operations

𝑅𝑜𝑑𝑃𝑆𝑇 The set of routes connecting 𝑜𝑑 when a PST is purchased

𝑅𝑜𝑑𝐹𝐴𝑇 The set of routes connecting 𝑜𝑑 when a FAT is purchased (𝑅𝑜𝑑

𝑃𝑆𝑇 ⊂ 𝑅𝑜𝑑𝐹𝐴𝑇)

𝑈 Coarse-scale time horizon

𝐴 Set of airspaces

𝐶𝑎, 𝑆𝑎 Set of configurations and elementary sectors for airspace 𝑎

𝑃𝑐 Partition of elementary sectors corresponding to a configuration

𝑆𝑝 Subset of elementary sectors forming the collapsed sector within a configuration

𝑂𝐿 Subset of origin-destination pairs used by late-booking flights

𝐹𝐿 The set of late-booking flights (booking on the day of operations)

Indices:

𝑓 Flights

𝑜𝑑 Origin and destination airports

EDITION [01.00.00]

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𝑢 Coarse-scale time index

𝑟 Route

𝑎 Airspace

𝑐, 𝑐′ Airspace’s configuration

𝑝 Airspace portion

𝑠 Elementary sector

Parameters:

𝑣𝑓 Charge weight dependent on the type of aircraft used by 𝑓

𝜌𝑎 Variable cost of providing one sector-time unit for airspace 𝑎

𝑘𝑝 Maximum capacity of airspace portion 𝑝

𝑞𝑎 Fixed cost of airspace 𝑎

ℎ̅𝑎𝑐 Number of sector hours consumed by airspace 𝑎 working in configuration 𝑐

�̅� Length (min) of a coarse-scale time unit

𝑑𝑟𝑓

Displacement cost of route 𝑟 for flight 𝑓

𝛿𝑟 Ground delay for route 𝑟

𝑡𝑓𝑜𝑓𝑓

Flight 𝑓 scheduled take off time

𝑏𝑟𝑠𝑢(𝑡𝑓𝑜𝑓𝑓)

Is equal to 1 if route 𝑟 uses sector 𝑠 at time 𝑢, assuming take off

𝑡𝑓𝑜𝑓𝑓

, 0 otherwise

𝑃𝑓𝑃𝑆𝑇 Is the probability of flight 𝑓 purchasing a PST

𝑃𝑓𝐹𝐴𝑇 Is the probability of flight 𝑓 purchasing a FAT (𝑃𝑓

𝐹𝐴𝑇 = 1 − 𝑃𝑓𝑃𝑆𝑇)

𝑝𝑜𝑑𝐿 Is the probability that a late-booking flight will connect pair 𝑜𝑑

𝐹(𝒀𝑃𝑆𝑇, 𝒀𝐹𝐴𝑇)

Is the set of flights that have not purchased a trajectory,formally defined as

{𝑓 ∈ 𝐹 | 𝑌𝑓𝑃𝑆𝑇 = 0 AND 𝑌𝑓

𝐹𝐴𝑇 = 0}

Variables:



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𝑔𝑜𝑑𝑡𝑃𝑆𝑇 Is the PST charge for airport pair 𝑜𝑑 at time t

𝑔𝑜𝑑𝑡𝐹𝐴𝑇 Is the FAT charge for airport pair 𝑜𝑑 at time t

𝑧𝑎𝑐𝑢 = {1, if airspace𝑎 uses configuration 𝑐 at time 𝑢0, otherwise

𝑦𝑓𝑟𝑃𝑆𝑇 = {

1, if flight𝑓 is assigned to route 𝑟 when a PST is purchased0, otherwise

𝑦𝑓𝑟𝐹𝐴𝑇 = {

1, if flight𝑓 is assigned to route 𝑟 when a FAT is purchased0, otherwise

𝑦𝑓𝑜𝑑 𝑟𝐿 = {

1, if late-booking flight 𝑓 is assigned to route 𝑟 when flying airport pair 𝑜𝑑 0, otherwise

With the above notation, we can formulate a Dynamic Program to model COCTA environment. The so-called value function 𝑉𝑡(𝑠𝑡𝑎𝑡𝑒) represents the revenue the NM is expected to collect from 𝑡 to the end of the booking horizon:

𝑉𝑡 (𝒀𝑷𝑺𝑻, 𝒀𝑭𝑨𝑻,

𝒈𝒕−𝟏𝑷𝑺𝑻, 𝑃

) =

𝐦𝐢𝐧𝑔𝒕𝑷𝑺𝑻≥𝑔𝒕−𝟏

𝑷𝑺𝑻,

𝒈𝒕𝑭𝑨𝑻≤𝑔𝒕

𝑷𝑺𝑻

∑ ∑ 𝜆𝑓𝑡′

𝒕

𝒕′=𝟏

(𝑃𝑓𝑃𝑆𝑇(𝒈𝑡

𝑃𝑆𝑇 , 𝒈𝒕𝑭𝑨𝑻) (𝑣𝑓𝑔𝑜𝑑𝑓

𝑃𝑆𝑇

𝒇∈𝐹(𝒀𝑃𝑆𝑇,𝒀𝐹𝐴𝑇)

− ∆𝑃𝑆𝑇𝑉𝑡+1(𝒀𝑷𝑺𝑻, 𝒀𝑭𝑨𝑻, 𝒈𝒕

𝑷𝑺𝑻, 𝑃))

+ 𝑃𝑓𝐹𝐴𝑇(𝒈𝑡

𝑃𝑆𝑇 , 𝒈𝒕𝑭𝑨𝑻) (𝑣𝑓𝑔𝑜𝑑𝑓

𝐹𝐴𝑇

− ∆𝐹𝐴𝑇𝑉𝑡+1(𝒀𝑷𝑺𝑻, 𝒀𝑭𝑨𝑻, 𝒈𝒕

𝑷𝑺𝑻, 𝑃)))

+ 𝑉𝑡+1(𝒀𝑷𝑺𝑻, 𝒀𝑭𝑨𝑻, 𝒈𝒕

𝑷𝑺𝑻, 𝑃),

with the following boundary condition at time period T+1:

for 𝑡= 1,… , 𝑇

(3)

𝑉𝑇+1 (𝒀𝑷𝑺𝑻, 𝒀𝑭𝑨𝑻,

𝒈𝑇𝑷𝑺𝑻, 𝑃

)=

𝐦𝐢𝐧𝒚,𝒛,

𝑔𝑻+𝟏𝑭𝑨𝑻≥𝑔𝑻

𝑷𝑺𝑻

∑( ∑ 𝑑𝑟𝑓

𝑟∈𝑅𝑜𝑑𝑓𝑃𝑆𝑇

𝑦𝑓𝑟𝑃𝑆𝑇 + ∑ 𝑑𝑟

𝑓

𝑟∈𝑅𝑜𝑑𝑓𝐹𝐴𝑇

𝑦𝑓𝑟𝐹𝐴𝑇)

𝒇∈𝑭

+ ∑ ∑ 𝑝𝑜𝑑𝐿

𝑜𝑑∈𝑂𝐿

( ∑ 𝑑𝑟𝑓

𝑟∈𝑅𝑜𝑑𝐹𝐴𝑇

𝑦𝑓𝑜𝑑 𝑟𝐿

𝑓∈𝐹𝐿

+ 𝑣𝑓𝑔𝑜𝑑 𝑇+1𝐹𝐴𝑇 )

for 𝑡= 𝑇 + 1

(4)

EDITION [01.00.00]

14


s. t. ∑ 𝑦𝑓𝑟

𝑃𝑆𝑇

𝑟∈𝑅𝑜𝑑𝑓𝑃𝑆𝑇

= 𝑌𝑓𝑃𝑆𝑇

∀𝑓 ∈ 𝐹 (5)

∑ 𝑦𝑓𝑟

𝐹𝐴𝑇

𝑟∈𝑅𝑜𝑑𝑓𝐹𝐴𝑇

= 𝑌𝑓𝐹𝐴𝑇

∀𝑓 ∈ 𝐹 (6)

∑ 𝑦𝑓𝑜𝑑 𝑟

𝐿

𝑟∈𝑅𝑜𝑑𝐹𝐴𝑇

= 1 ∀𝑓 ∈ 𝐹

∀𝑜𝑑∈ 𝑂𝐿

(7)

∑( ∑ ∑ 𝑏𝑟𝑠𝑢(𝑡𝑓𝑜𝑓𝑓

+ 𝛿𝑟) 𝑦𝑓𝑟𝑃𝑆𝑇

𝑠∈𝑆𝑝𝑟∈𝑅𝑜𝑑𝑓𝑃𝑆𝑇𝑓∈𝐹

+ ∑ ∑ 𝑏𝑟𝑠𝑢(𝑡𝑓𝑜𝑓𝑓

+ 𝛿𝑟) 𝑦𝑓𝑟𝐹𝐴𝑇

𝑠∈𝑆𝑝𝑟∈𝑅𝑜𝑑𝑓𝐹𝐴𝑇

)

+ ∑ ∑ 𝑝𝑜𝑑𝐿

𝑜𝑑∈𝑂𝐿

∑ ∑ 𝑏𝑟𝑠𝑢(𝑡𝑜𝑓𝑠∈𝑆𝑝𝑟∈𝑅𝑜𝑑𝑓

𝐹𝐴𝑇𝑓∈𝐹𝐿

+ 𝑔𝑑𝑟) 𝑦𝑓𝑟𝑜 𝑑𝐿 ≤ 𝐾𝑝𝑧𝑎𝑐𝑢 + |𝐹| ∑ 𝑧𝑎𝑐′𝑢

𝑐′≠𝑐

∀𝑎 ∈ 𝐴,

𝑐 ∈ 𝐶𝑎,

𝑝 ∈ 𝑃𝑐 ,

𝑢 ∈ 𝑈

(8)

𝑃 + ∑ ∑ 𝑝𝑜𝑑𝐿

𝑜𝑑∈𝑂𝐿

𝑣𝑓𝑔𝑜𝑑 𝑇+1𝐹𝐴𝑇

𝑓∈𝐹𝐿

≥ ∑(𝑞𝑎𝑎∈𝐴

+ 𝜌𝑎∑ ∑ ℎ̅𝑎𝑐𝑧𝑎𝑐𝑢𝑐∈𝐶𝑎𝑢∈𝑈

) (9)

∑∑ℎ̅𝑎𝑐𝑧𝑎𝑐𝑢𝑐∈𝐶𝑢∈𝑈

≤ ℎ𝑎 ∀𝑎 ∈ 𝐴 (10)

𝑦𝑓𝑟𝑃𝑆𝑇 , 𝑦𝑓𝑟

𝐹𝐴𝑇 ∈ {0, 1} ∀𝑓 ∈ 𝐹

𝑟 ∈ 𝑅𝑜𝑑𝑓 (11)

𝑦𝑓𝑜𝑑𝑟𝐿 ∈ {0, 1}

∀𝑓 ∈ 𝐹

∀𝑜𝑑∈ 𝑂𝐿

𝑟 ∈ 𝑅𝑜𝑑

(12)

𝑧𝑎𝑐𝑢 ∈ {0, 1}

∀𝑎 ∈ 𝐴,

𝑐 ∈ 𝐶𝑎,

𝑢 ∈ 𝑈

(13)



15

The recursive function 𝑉( ) is defined in (3) for 𝑡 ≤ 𝑇. Its form is very similar to FPP, with the difference that only two types of flexible products are considered and price decisions of these products are intertwined. The marginal costs are defined as follows:

∆𝑃𝑆𝑇𝑉𝑡+1( ) = 𝑉𝑡+1 (𝒀𝑷𝑺𝑻 + 𝟏𝒇, 𝒀

𝑭𝑨𝑻, 𝒈𝒕𝑷𝑺𝑻, 𝑃 + 𝑣𝑓𝑔𝑜𝑑𝑓𝑡

𝑃𝑆𝑇 ) − 𝑉𝑡+1(𝒀𝑷𝑺𝑻, 𝒀𝑭𝑨𝑻, 𝒈𝒕

𝑷𝑺𝑻, 𝑃)

∆𝐹𝐴𝑇𝑉𝑡+1( ) = 𝑉𝑡+1 (𝒀𝑭𝑨𝑻, 𝒀𝑭𝑨𝑻 + 𝟏𝒇, 𝒈𝒕

𝑷𝑺𝑻, 𝑃 + 𝑣𝑓𝑔𝑜𝑑𝑓𝑡𝐹𝐴𝑇 ) − 𝑉𝑡+1(𝒀

𝑷𝑺𝑻, 𝒀𝑭𝑨𝑻, 𝒈𝒕𝑷𝑺𝑻, 𝑃).

The boundary condition (4-13) on the value function at time 𝑇 + 1, is more complex. The objective (4) at time 𝑇 + 1 is to find the optimal assignments which minimise overall costs (capacity cost - charges and displacement). At this stage we assume that more of 90% of the demand is known and has purchased a product. The rest of the demand is uncertain and is estimated through probability distributions. Constraints (5)-(7) enforce that all flights which already purchased PST and FAT as well as all flights expected to show up later are assigned to a trajectory. Constraints (8) is the capacity constraint for each sector. It is formulated in the same way as in the D5.1. Inequality (9) state that the total revenue collected plus the revenue collected from late bookings (𝑃 +

∑ ∑ 𝑝𝑜𝑑𝐿

𝑜𝑑∈𝑂𝐿 𝑣𝑓𝑔𝑜𝑑 𝑇+1𝐹𝐴𝑇 )𝑓∈𝐹𝐿 must fully cover the capacity provision costs. Constraints (11) are the

sector-hours budget constraints for each airspace. Finally, constraints (12)-(13) defines the feasibility sets for the decision variables.

EDITION [01.00.00]

16


3 Solution Approach

We discuss the solution approach on the example of the simpler, generic problem (1)-(2) so as to improve readability of the document. If we wanted to solve it to optimality, the value function must be evaluated for each state and time period. Since the number of states is huge, a complete enumeration is possible only for very small problem instances. In fact, the number of states grows exponentially with the number of customers and flexible products. Intractable dynamic programming problems due to large state spaces are well-known in many application areas, as discussed in the book of Powell (2007). Various solution approaches for such problems in the context of choice-based demand management have been proposed, including affine (Zhang and Adelman 2009) and piecewise-linear approximations (Kunnumkal and Talluri 2016). These approximation approaches reduce the problem to a deterministic linear program, the size of which can be reduced using recent results of Vossen and Zhang (2015).

In this section, we propose a version of such a linear programming approximation, i.e. Deterministic Linear Programming (DLP). We assume that demand is deterministic and equal to its expected values.

Parameter 𝑌𝑖𝑙 denotes the number of requests of product 𝑖 from segment 𝑙 accumulated from the

beginning to 𝑡. Barred assignment variables �̅� are used to account for past purchases, whereas non-barred 𝑦 are the assignment variables for the expected demand-to-come.

[DLP(𝒀, 𝑡)]

𝐦𝐚𝐱𝒙𝒈

∑∑∑(𝑔𝑖𝜆𝑙𝑃𝒊𝒍(𝒈)𝑥𝑔𝑖(𝑇 − 𝑡) − ∑ 𝑦𝑖𝑗

𝑙

𝒋∈𝑹𝒊

𝑑𝑗)

𝑖∈𝐼𝑙𝒈𝑙

−∑∑∑𝑑𝑗 �̅�𝑖𝑗𝑙

𝒋∈𝑹𝒊𝑖∈𝐼𝑙𝑙

(14)

s. t. ∑∑ 𝑎𝑗𝑚 (𝑦𝑖𝑗𝑙 + �̅�𝑖𝑗

𝑙 ) ≤ 𝐵𝑚

𝒋∈𝑹𝒊𝒊∈𝐼𝑙

∀𝑚 (15)

∑�̅�𝑖𝑗𝑙 = 𝑌𝑖

𝑙

𝒋∈𝑹𝒊

∀𝑙, 𝑖 (16)

∑𝑦𝑖𝑗𝑙

𝒋∈𝑹𝒊

= 𝜆𝑙∑𝑃𝒊𝒍(𝒈)

𝑔

𝑥𝑔𝑙 (𝑇 − 𝑡) ∀𝑙, 𝑖 (17)

∑𝑥𝒈 = 1

𝒈

(18)

𝑦𝑖𝑗𝑙 , �̅�𝑖𝑗

𝑙 ≥ 0 ∀𝑖, 𝑗 (19)

𝑥𝑔𝑙 ∈ [0, 1] ∀𝑔, 𝑙 (20)

The objective (14) is to maximize the total expected revenue from time 𝑡 to the end of the booking horizon (NM’s revenue neutrality is preserved). Constraints (15) are the budget limitations for each



17

resource. Constraints (16) state that the entire “appeared” demand on product 𝑖 must be assigned to a specific product. Constraints (17) define a portion of expected demand to assign to specific products. Equality (18) enforce that only one price vector can be selected. Finally, constraints (18) and (19) are the binary requirements for variables 𝑦 and 𝑥. Charge variables are segment indexed because we work on the assumption that segments are non-overlapping (customers of different segments are offered different products).

The idea behind the solution approach is to solve DLP a few times during the booking horizon. Each subsequent time the model will have a clearer picture of what is the portion of demand which already appeared and what to expect from the future. The output of the model will be a pricing distribution which can be sampled to identify prices to adopt until the next re-solving of the model.

In Figure 3-1 there is a graphical representation of the solution approach. The DLP is solved at the

beginning of the booking horizon, entirely based on expectation (𝒀 is the null vector, therefore �̅�𝑖𝑗𝑙

variables are all set to zero). Optimal variables 𝑥𝒈 are then used to build a price policy. This policy

remains unchanged until time 𝑡1, when the DLP is solved again (now vector 𝒀 is no longer null and stores the number of purchases made by customer up to 𝑡1). The new solution is used to eventually update products' prices. DLP is re-solved for a number of predefined times until the end of the booking horizon.

Once the sale is over, a DLP is further solved with the entire demand known to identify the optimal flexible-specific products assignments. In the remainder of this section, we will perform a computational analysis to evaluate the scalability of this approach.

3.1 Small-scale example

In this subsection, a small example is built to provide a better understanding of the algorithm in action. We consider 10 available specific products. These products are built combining units of 11 resources. The incidence matrix explaining the product-resource consumption is given in Table 3-1. The table also lists a “dummy” product (𝑦0) using a “dummy” resource (𝑚0) (highlighted in red). This product is added to avoid unfeasible solutions. The “dummy” resource is assumed to have no budget constraint.

Figure 3-1 Illustration of the solution approach

EDITION [01.00.00]

18


The first four products use only one unit of one resource. The reader can think of these as the shortest routes in the ATM context. Other products use 2, 3 and 4 different resources and therefore can be interpreted as longer routes.

The production cost of a specific product directly translates into displacement costs as described in COCTA. They are both specific-products costs which are subtracted from the firm/NM objective and, therefore, the aim is to minimize them. With the ATM problem in mind, cost of production have been set to replicate the displacement costs. For instance, no production cost is incurred with one resource products (i.e., no displacement cost for shortest routes). The production costs gradually increase as the number of resources increases. In other words, products which require more resources (longer trajectories) are more expensive for the firm’s (NM’s) point of view. Finally, a significantly higher cost is linked with the dummy product, to simulate the unlikely situation in which the firm cannot deliver a product that has already been sold out. Production costs are shown in Table 3-2.

Table 3-1 Resource consumption of specific products

Resources Dummy

𝑚1 𝑚2 𝑚3 𝑚4 𝑚5 𝑚6 𝑚7 𝑚8 𝑚9 𝑚10 𝑚11 𝑚0

Specific P

rod

ucts

𝑗1 1

𝑗2 1

𝑗3 1

𝑗4 1

𝑗5 1 1

𝑗6 1 1

𝑗7 1 1 1

𝑗8 1 1 1

𝑗9 1 1 1 1

𝑗10 1 1 1 1

Dummy 𝑗0 1

Table 3-2 Production costs of specific products

Specific products (Production cost) 𝑦0

0 0 0 0 2 2 4 4 6 6 20

We consider six different customer segments. Each segment has access to two different flexible products. Flexible products have also been designed with reference to the COCTA problem. In fact, each segment has one flexible product containing only the options with smaller production costs (1



19

and a 2-resource products). For example, 𝑖1 can be assigned to either 1-resource products 𝑗1 and 𝑗2 (with no production cost) or to 2-resource product 𝑗5 (with low production cost). This should well replicate the PST where only small displacements are considered. The FAT option is considered with the second flexible product which has also 3 and 4-resources products, leading to higher production costs.

The definition of flexible products and their assignments to segments are explained in Tables 3-3 and 3-4.

Each segment has a known rate of arrival. We assume that at any time period there is one arrival (i.e., ∑ 𝜆𝑙𝑙 = 1). Therefore, the length of the booking horizon 𝑇 can also be interpreted as the total number of customer expected to arrive. In the example, we set 𝑇 = 1000. The rates of arrival are shown in Table 3-5.

Table 3-3 Flexible to specific products definition

Specific products

Flexible p

rod

ucts

𝑖1 𝑗1 𝑗2 𝑗5 𝑗0

𝑖2 𝑗1 𝑗3 𝑗6 𝑗0

𝑖3 𝑗1 𝑗4 𝑗5 𝑗0

𝑖4 𝑗2 𝑗3 𝑗6 𝑗0

𝑖5 𝑗2 𝑗4 𝑗5 𝑗0

𝑖6 𝑗3 𝑗4 𝑗6 𝑗0

𝑖7 𝑗1 𝑗2 𝑗5 𝑗7 𝑗9 𝑗0

𝑖8 𝑗1 𝑗3 𝑗5 𝑗7 𝑗9 𝑗0

𝑖9 𝑗1 𝑗4 𝑗5 𝑗7 𝑗9 𝑗0

𝑖10 𝑗2 𝑗3 𝑗6 𝑗8 𝑗9 𝑗0

𝑖11 𝑗2 𝑗4 𝑗6 𝑗8 𝑗10 𝑗0

𝑖11 𝑗3 𝑗4 𝑗6 𝑗8 𝑗10 𝑗0

EDITION [01.00.00]

20


Table 3-4 Flexible products available per segment

Flexible products

Cu

stom

er segm

ents

𝑙1 𝑖1 𝑖7

𝑙2 𝑖2 𝑖8

𝑙3 𝑖3 𝑖9

𝑙4 𝑖4 𝑖10

𝑙5 𝑖5 𝑖11

𝑙6 𝑖6 𝑖11

Table 3-5 Customer rate of arrival per segment

Rate of arrival per segment (𝜆𝑙)

0.17 0.17 0.17 0.17 0.16 0.16

To compute the choice probabilities, we used one of the most common choice model, the Multinomial Logit model (MNL). With MNL, the probability of a customer from segment 𝑙 of purchasing product 𝑖, given prices 𝑔, is computed as follows:

𝑃𝒊𝒍(𝒈) =

𝑣(𝑢𝑖 − 𝑔𝑖)

∑ 𝑣(𝑢𝑘 − 𝑔𝑘)𝑘∈𝐼𝑙

where 𝑢𝑖 represents a quantitative evaluation of how important product 𝑖 is to the customer (see Table 3-6).

Table 3-6 Customer evaluation of flexible products

Flexible products

Cu

stom

er segm

ents

𝑙1 𝑖1(10) 𝑖7(6)

𝑙2 𝑖2(10) 𝑖8(9)

𝑙3 𝑖3(10) 𝑖9(6)

𝑙4 𝑖4(10) 𝑖10(9)

𝑙5 𝑖5(10) 𝑖11(6)

𝑙6 𝑖6(10) 𝑖11(9)

As the table indicates, we modelled two types of behaviour (10-6 and 10-9): the first simulates customers who value 1 and 2 resources product significantly more over other products (10 vs 6) and are prepared to pay more for that. In COCTA, this would apply to AO who have strict time constraints and are strongly motivated to pay more for a PST. The second behaviour shows a smaller difference in the evaluation of the two flexible products (10 vs 9). This simulates customers who still prefer the less flexible product (for obvious reasons) but are not equally willing to pay much more for it. Again, in the COCTA context this translates into those AOs who are willing to take advantage of the discounts offered along with FAT products.



21

We consider a discrete set of prices to choose from. Each product can be offered at one of the prices listed in Table 3-7.

Table 3-7 Discrete price set

Set of prices (𝒈)

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 𝑔0

Last price 𝑔0 is called the null price because 𝑃𝒊𝒍(𝒈) = 0 if 𝑔𝑖 = 𝑔0. In other words, offering a product

at the null price is equivalent to not offering the product at all.

Finally, capacity limitations on resources are chosen to obtain a challenging, yet feasible example. We purposely set tight limitations on the first six resources (the ones used to create the most attractive products), while leaving no limitations on the others. In the ATM context, this can be seen as observing a bottleneck over a subset of airspaces or sectors which are part of several shortest routes, while other neighbouring sectors have relatively low traffic. Resource capacity budget are shown in Table 3-8.

Table 3-8 Budget available per resource

Resource budget (𝐵𝑚)

𝑚1 𝑚2 𝑚3 𝑚4 𝑚5 𝑚6 𝑚7 𝑚8 𝑚9 𝑚10 𝑚11 𝑚0

50 50 50 100 50 100 ∞ ∞ ∞ ∞ ∞ ∞

With this given product-resource network, we perform a simulation to evaluate the impact of the algorithm introduced above. The simulation consists into sampling customer arrivals and their product choices over the booking horizon, while offering products at the prices suggested by the DLP. The analysis is iterated for 1000 times and average results were obtained as output.

The first test is aimed at assessing the impact on the solutions of the number of times the DLP is solved. The benchmark of the solution is the DLP itself solved based on the expectation of the demand over the entire time horizon (i.e., DLP(𝟎, 𝑇)). Deterministic linear programs have been proven to represent upper bounds on the optimal objective value of the underpinning optimal control problem. Empirical results suggest that this is the case for our problem as well. The mathematical proof of the upper bound condition is beyond the scope of this deliverable. We cannot make any estimation on how tight the bound is. Nonetheless, the DLP still provides a valuable information. In fact, given that our algorithm will return a lower bound, if it is close enough to the upper bound then it means that it is close to the optimal value as well. Conversely, a big gap between the two bounds does not necessarily mean a bad solution from the algorithm.

In Figure 3-2, the estimated upper bound (UB) is highlighted in dark blue. The light blue line shows the revenue returned by our algorithm depending on the number of times DLP is solved. The percentage gap between algorithm and UB is shown by the green line. The graph shows that by solving DLP only once upfront, the result is already close to the UB (less than 4% difference). The beneficial impact of

EDITION [01.00.00]

22


resolving the DLP during the booking horizon is clear. Solving DLP two times reduces the gap to less than 2% and only 5 solutions are needed to have a gap smaller than 1%.

Figure 3-1 Solution approach vs estimated upper bound

In the following graph (Figure 3-3), we provide insight on how the price of one product may change during the booking horizon. Specifically, we focus on flexible product 𝑖7. With a single solution approach (1-Sol), DLP is solved once upfront and the price of the product is set to 10 and does not change. On the other hand, if we solve DLP 10 times (10-Sol), the price will be firstly set to 10, then reduced to 9.5 for most of the time and by the end of the horizon will be increased to 10 again, before finally going back to 9.5. In COCTA we consider prices to be either constant or increasing. Nonetheless, the amount of discount between FAT and PST can change without constraints. Therefore, we will see a similar behaviour to the one shown in the graph, with PST discount changing across the booking horizon as more AO purchase trajectories and the NM acquires more information.

Figure 3-3 Example of price decisions on flexible product 𝑖7

Similar price variations can be observed on all the products. For sake of brevity, results are omitted.

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

4300

4350

4400

4450

4500

4550

4600

4650

1 2 5 10 50 100

Rev

enu

e d

iffe

ren

ce

Rev

enu

e

Number of times DLP is solved

Algorithm Upper Bound Difference(%)

6

6.5

7

7.5

8

8.5

9

9.5

10

10.5

0 100 200 300 400 500 600 700 800 900

Pro

du

ct p

rice

𝑖_7

Booking horizon

1-Sol

10-Sol



23

3.2 Large-scale networks

One of the biggest issues within the COCTA problem is its complexity in terms of size. On an average day about 28,000 flights use the European airspace (see, e.g. EUROCONTROL PRC, 2017). Even if we narrow the focus to only the congested area, the numbers are still high. A solution algorithm must be able to work on a network of similar dimension. In this section, we analyse how well the proposed algorithm scales up with the size of the problem.

In COCTA each flight translates into a segment of the RM problem. Consequently, we aim to test the scalability of our approach in function of the number of segments. To this aim, we keep the same product-resource network while scaling up the problem by replicating the six segments introduced above. Let us define 𝜃 as the segment multiplier, each network instance will have 6 ∗ 𝜃 segments. For example, with 𝜃 = 2 the segments will be defined as shown in Table 3-9:

Table 3-9 Segments definition when 𝜃 = 2

𝜃 = 2 Flexible products

Cu

stom

er segm

ents

𝑙1 𝑖1(10) 𝑖7(6)

𝑙2 𝑖2(10) 𝑖8(9)

𝑙3 𝑖3(10) 𝑖9(6)

𝑙4 𝑖4(10) 𝑖10(9)

𝑙5 𝑖5(10) 𝑖11(6)

𝑙6 𝑖6(10) 𝑖11(9)

𝑙7 𝑖1(10) 𝑖7(6)

𝑙8 𝑖2(10) 𝑖8(9)

𝑙9 𝑖3(10) 𝑖9(6)

𝑙10 𝑖4(10) 𝑖10(9)

𝑙11 𝑖5(10) 𝑖11(6)

𝑙12 𝑖6(10) 𝑖11(9)

With this approach we can easily test large realistic networks to replicate the COCTA environment. Despite replicating segments and their flexible products associations, we still operate under the assumption that no overlap is possible among segments. The scaling parameter 𝜃 is also used to increase the available budget on each resource.

Aside from varying the number of segments, we also test a scenario with 25 price points (24 + the null price). Results of exponentially increasing the number of segments are reported both in Figure 3-4 and Table 3-10.

EDITION [01.00.00]

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Figure 3-4 Computational time of solving DLP - illustration

Table 3-20 Computational time of solving DLP - Data

Number of

segments

Scaling parameter

θ

Computational time - 12+1

price points(s)

Computational time - 24+1

price points(s)

30 5 0.33 1.52

60 10 0.67 2.95

120 20 1.35 6.36

240 40 2.44 15.27

480 80 5.09 28.61

960 160 9.20 42.42

1920 320 22.50 91.75

3840 640 57.88 181.47

7680 1280 103.96 466.25

Results show that the problem appears to be scalable with the number of segments. In fact, solving the model for 7680 segments takes 103.96 seconds. Considering that the solution approach proposed requires to solve the DLP for a limited number of times, these numbers prove its applicability to problems of realistic sizes. Incrementing the number of price points has a clear impact on the DLP complexity. For instance, solving the more complex instance requires 466.25s. Nonetheless, the number of price points considered in the example is likely to be enough for our problem. In fact, in COCTA it is realistic to assume that only a finite number of possible FAT discounts will be considered. Therefore, working with up to 24 different discounts is a realistic assumption.

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30 60 120 240 480 960 1920 3840 7680

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To further discuss scalability, we provide an on-scale graph (Figure 3-5) with the first 5 rows of Table 3-10. We focus on a subset of results solely for the sake of readability. The graph clearly shows that computational time grows almost linearly with the number of segments.

Figure 3-5 Scaled graph of computational time

0

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Linear (12+1 price points) Linear (24+1 price points)

EDITION [01.00.00]

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4 COCTA Implications

In this section, we recap contributions and results of this deliverable highlighting the implication to COCTA mechanism.

Section 2 focuses on the mathematical formulation. Specifically, in Section 2.1, we introduce a generic revenue management optimization model. This problem bears large similarities with the COCTA mechanism.

In sub-sections 2.1.x, we reflect on these similarities while also highlighting a few differences. This leads to a COCTA-specific formulation of the problem, based on the same structure of the RM model. Unfortunately, neither the RM nor the COCTA problem can be easily solved to optimality with large problem instances. Nonetheless, the similar mathematical structure allows us to use for COCTA same approximate solution techniques which have been widely studied and implemented in RM context.

Section 3 is devoted to introducing and evaluating the chosen solution algorithm. Given the lack of data to perform a large scale computational experiment, we use the RM problem, confident that results can be transferred to COCTA context. The algorithm’s core is a Deterministic Linear Program which approximates the full dynamic and stochastic problem. This DLP is repeatedly solved during the booking horizon, providing new pricing decisions based on the revealed and expected demand and its impact on resources budget. In practical terms, the Network Manager will solve this DLP a few times to adjust FAT and PST prices based on his knowledge of flights which have purchased or are expected to purchase a product and the impact that the consequent route assignments will have on airspace configurations.

In Section 3.1, we report the results of the analysis on a small-scale network. The purpose of this analysis is to test the precision of the solution approach. Figure 3-2 indicates that during the 6 months of trajectories selling, the NM might need to update its price only a few times and only when substantial changes in demand happen. This reduces the computational overhead without trading off with algorithms precision. It is a critical advantage, in light of the geographical scale and rolling horizon feature of the COCTA problem. With this solution approach, the NM will effectively have to solve only a small number of DLPs every day.

Price changes in Figure 3.3 show that the algorithm reacts to changes in its resource capacities and reflect these changes in the dynamic prices. Applied to COCTA, this means that by re-solving a few times, the deterministic linear program will give enough input to the NM which, in turn, will adjust prices for its FAT and PST products based on the revealing traffic over the airspaces.

For example, Figure 3.3 shows the price of a flexible product with wider margins (i.e., including also specific products made of 3 and 4 resources). As discussed earlier, this product could be interpreted as a FAT. Dynamic pricing in blue would suggest that for most of the booking horizon the NM encourages FAT purchases by reducing its price (compared to the initial price of 10). The graph also shows how the NM could change price toward the end as the late uncertain demand appears. In fact, the price of the FAT product is firstly increased then decreased again. This is a clear example of how



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the NM could carefully make a FAT product more or less attractive to gain the trajectory assignment flexibility necessary to deal with the final tranche of demand appearing.

With the large-scale analysis the focus is shifted to the scalability of the approach. More specifically, the aim is to evaluate how realistic it will be for the NM to re-compute the DLP a few times each day. Computational results suggest that working with thousands of flights prompts computational times in the range of minutes (see Table 3-10). Given that we expect the NM to solve the DLP roughly 10 times a day on average, results are promising and also leave plenty of room to tackle other COCTA features that have been set aside in the computational experience (e.g., decisions on sector configurations).Conclusions and next steps

In this deliverable, we have provided a formulation to model pricing and flexible trajectory products and a solution approach to tackle such complex problem. The algorithm has been tested with a generic Revenue Management network, showing good results both in terms of accuracy and efficiency.

As we already highlighted, COCTA’s trajectory selling problem bears many similarities with the generic Revenue Management problem used as benchmark. The main critical difference is the way capacity is incorporated. In fact, in RM resources have static budget limitations, whereas in COCTA sectors capacities are driven by airspace configurations. The number of configurations can be a factor affecting the computational time. A set of pre-computational traffic-driven rules will be developed to assess whether some configurations will be unlikely to be used at a specific time. With this information, we will be able to actively reduce the number of combinations and therefore reduce the number of decision variables and constraints.

We also expect the NM to have a good estimation (based on past observations) on when a scheduled flight is likely to book a trajectory. Focusing on the entire booking horizon, it is reasonable to assume that for a given flight 𝑓, its rate of arrival 𝜆𝑓𝑡 will be non-zero only for a small subset of time periods 𝑡.

This information can be very useful to reduce the number of flights considered as we advance in the booking process.

Ultimately, it is important to point out that the Revenue Management literature has heavily focused on finding efficient approaches to solve DLPs. We are planning to draw on these results to tackle the additional expected computational challenges.

In the next deliverables, we will consider the possibility of ordering some additional airspace capacity. We will focus on how this issue can be incorporated in our algorithmic framework. The main challenge is to harmonize decisions made at different time scales (daily vs annual). We are already testing different solutions, one of which is the key idea the paper COCTA team submitted for the SESAR Innovation Days 2017 conference paper. Also, the COCTA team will develop a rule-based choice model, following up on the consultation with AO representatives (see D4.2).

The algorithm will be further refined so that it performs efficiently in the COCTA context. We will test this approach with a large-scale case study based on real data of the usage of European airspace. This case study will be used to show the practicality and advantage of the COCTA ideas.

EDITION [01.00.00]

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5 References

[1] COCTA consortium, 2017. Prototype models and small academic examples.

[2] EUROCONTROL PRC, 2017. Performance review report 2016.

[3] Kunnumkal, S., Talluri, K., 2016. On a piecewise-linear approximation for network revenue management. Mathematics of Operations Research 41 (1), 72-91.

[4] Petrick, A., Steinhardt, C., Gönsch, J., & Klein, R. (2012). Using flexible products to cope with demand uncertainty in revenue management. OR Spectrum, 34(1), 215-242.

[5] Powell, W., 2007. Approximate dynamic programming, 2nd Edition. Wiley, Hoboken

[6] Starita, S., Strauss, A., Jovanovic, R., Ivanov, N., Fichert, F., 2016. Maximizing ATM Cost-efficiency by Flexible Provision of Airspace Capacity, in: SESAR Innovation Days (SIDs 2016). Delft.

[7] Zhang, D., Adelman, D., 2009. An approximate dynamic programming approach to network revenue management with customer choice. Transportation Science 43 (3), 381-394.



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