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IN

DEGREE PROJECT MATHEMATICS, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2019,

Multi-Objective Mixed-Integer Linear Optimisation of Aircraft Load Planning

KRISTJÁN ÓTTAR RÖGNVALDSSON

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

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Multi-Objective Mixed-Integer Linear Optimisation of Aircraft Load Planning

KRISTJÁN ÓTTAR RÖGNVALDSSON

Degree Projects in Optimization and Systems Theory (30 ECTS credits) Degree Programme in Aerospace Engineering (120 credits)

KTH Royal Institute of Technology year 2019

Swedish title: Flermåls Linjär Blandad-Heltalsoptimering av Flygplanslastning Supervisor at KTH: Anders Forsgren

Examiner at KTH: Anders Forsgren

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TRITA-SCI-GRU 2019:221 MAT-E 2019:62

Royal Institute of Technology School of Engineering Sciences KTH SCI

SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

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iii

Abstract

A general multi-objective optimisation model is developed for the load plan- ning decision process of a bulk loaded commercial aircraft, using the Airbus A321 fitted with additional fuel tanks as a baseline platform. The model’s in- put is a specific set of load items, with associated quantities, mass and volume.

The output is a load plan, stating where each item should be loaded and in what quantity. The load plans should be optimal with respect to a target centre of gravity range and handling efficiency. Furthermore, the solutions should be robust with respect to perturbations in the input data. Three objective func- tions and a set of constraints are defined to achieve this task. A constraint that ensures the ground stability of the aircraft is developed and analysed.

A lexicographic approach is used solve the multi-objective problem, by se- quentially solving a set of mixed-integer linear programs. The sequence is determined from a priority ranking of the objectives. Testing is carried out with data from an operator of the A321, with four different test cases.

Test results indicate that the model is capable of solving the load planning problem for the baseline aircraft. The centre of gravity values are within the optimal range, and the load distributions are efficient. Additional margins on aircraft limits assist with maintaining feasibility in case of input perturbation.

The model is also robust with respect to the highly variable test data. The main causes of infeasibility are mixing constraints and additional balance en- velope margins. The ground stability constraint does not cause any significant amount of infeasibilities, and primarily increases the safety level of the load plans. A strength of the model is its relatively simple handling of the multiple objectives, and the small number of tunable parameters also makes the model controllable. A trained agent in the industry is able to understand and control the model without an extensive technical background.

The test process used differs slightly from the actual industry load planning process. As a result, testing only allows for evaluation of the model’s ability to solve the load planning problem, and gives no justification for implementation in real-world operations. Such an evaluation requires a prototype to be tested in an operational environment using the actual process.

As testing was only done for the baseline aircraft, with one set of test data and model parameters, a justifiable conclusion cannot be reached on the model’s applicability to other bulk loaded aircraft. Therefore, it is recommended to carry out further testing on different aircraft as the next step in them model’s evaluation.

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iv

Sammanfattning

En allmänn flermåls optimeringsmodell utvecklas för besluttsprocessen rela- terat till lastning av ett kommersiellt flygplan, som använder Airbus A321 ut- rustad med ytterligare bränslestankar som en bas. Modellens indata är ett spe- cifikt set av artiklar som ska lastas tillsammans med information om mängd, tyngd och volym. Utvärdet är en beskrivande plan som visar var varje arti- kel ska lastas och i vilken mängd. Planen ska vara optimal med hänsyn till ett specifikt tyngdpunkts intervall och vara effectiv för lastningsoperationer.

Dessutom ska den vara robust med hänsyn till störningar i indatan. Tre mål- funktioner tillsammans med ett set av begränsningar används för att lösa pro- blemet. En specifik begränsning som säkrar flygplanets stabilitet på grunden utvecklas och dess känslighet analyseras. En lexikografisk metod används för att lösa flermåls problemet, där lösar en sekvens av blandade heltalsprogram- mer. Sekvensen är definierad ut från en prioritetsordning av de olika målfunk- tioner. Testning av modellen är utförd med indata från en operatör av A321 basflygplanet med fyra olika testfall.

Testresultaten visar att modellen kan användas för att lösa lastningsproble- met för basflygplanet. Tyngdpunktsvärden är inom det optimal intervall och fördelningen av artiklar är effektiv. Extra marginaler på flygplansbegränsning- ar hjälper med att säkra lösningen under störningar på indatan. Modellen är också robust med hänsyn till högvarierad indata. Huvudorsaker till omöjliga testfall, de utan lösningar, är begränsningar på blandning av artiklar samt ex- tra marginaler på flygplansbegränsningar. Begränsningen för grund stabilitet är inte en orsak till omöjlighet, och ökar primärt säkerhetsnivån på lösningen.

En styrka till modellen är dess enkel hantering av de olika målfunktioner och de få parametrar gör modellen kontrollbar. En utbildad agent från industrin kan förstå och kontrollera modellen, utan att ha en teknisk bakgrund.

Testprocessen som används representerar inte exakt industriprocessen. Test- processen kan därför bara användas till att utvärdera modellens förmåga till att lösa lastningsproblemet, och ger ingen motivering på bruk i verkliga ope- rationer. En utvärdering på den förmåga krävs en utveckling av en prototyp i verkliga världen.

Testning av bara en typ av basflygplan, tillsammans med ett set av indata och modellparametrar, ger inte en grund till en konklusion på modellens tillämp- lighet för andra flygplan. Därför rekommenderas det att utföra ytterligare test- ning på andra flygplan som nästa steg i modellens utvärdering.

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v

Acknowledgments

My relationship with the topic of aircraft load planning stems primarily from hands-on industry experience in the past 7 years; as a load controller tasked with creating load plans for various aircraft and operators, in the role of train- ing others and from designing and implementing such processes in an air- line environment. Therefore, great appreciation is extended to those that have trained me, shared knowledge and experience as well as collaborating throughout that time. A great bulk of this past experience has been used for this project.

Test data and other industry information required to develop and evaluate the model proposed was provided by my colleagues and friends at Res2 B.V. in the Netherlands. Without that contribution, specific model details and eval- uation would not be possible. My thanks are extended to them for their gra- ciousness and interest in assisting me with the project in any way possible.

My friends and fellow students in the Aerospace Master Program who provided a warm atmosphere in the bitter Stockholm winters, and chill in the summer heat deserve a special kind of thanks. They have introduced me to societies, languages and views from both far and near, and hopefully a connection for life. Debates, agreements, disagreements on course topics and other ideas that may fly in specific circumstances, gave me a learning experience and a different viewpoint on the world.

Finally, my family deserves heartfelt appreciation for all assistance in the last few years, for their visits to Stockholm during my time there, accommodation and assistance during my work on this project, and support in life.

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Contents

1 Introduction 1

1.1 Problem Description and Scope . . . 2

1.2 Project Goal . . . 5

1.3 Thesis Structure . . . 5

2 Literature Review 6 3 Methodology 8 3.1 Theoretical Framework . . . 8

3.1.1 Aircraft Weight and Balance . . . 8

3.1.2 Lexicographic Mixed Integer Linear Programming (LMILP) . . . 11

3.2 Model . . . 13

3.2.1 Index Sets . . . 14

3.2.2 Variables . . . 14

3.2.3 Model Constraints . . . 15

3.2.4 Aircraft Limits Constraints . . . 16

3.2.5 Operational Constraints . . . 21

3.2.6 Objectives . . . 23

3.3 Solution Process . . . 25

3.4 Testing . . . 26

3.4.1 Test Data . . . 26

3.4.2 Solvers & Hardware . . . 29

4 Results 30 4.1 Case 1 . . . 30

4.2 Case 2 . . . 32

4.3 Case 3 . . . 34

4.4 Case 4 . . . 36

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CONTENTS vii

5 Discussion 39

5.1 Load Plan Quality . . . 39

5.1.1 Centre of Gravity . . . 39

5.1.2 Compartment Selection . . . 40

5.1.3 Ground Stability Slack . . . 41

5.2 Causes of Infeasibility . . . 42

5.3 Solver Performance . . . 44

5.4 Test Validity . . . 44

6 Conclusions 46 Bibliography 48 A Ground Stability 50 A.1 Fuelling . . . 51

A.2 Passenger Movement . . . 52

A.3 Compartment Loading . . . 53

A.4 Ground Stability Constraint . . . 53

A.5 Bounding the Stability Margin . . . 55

A.6 Sensitivity Analysis . . . 57

B Baseline Optimisation Model 59 B.1 Model Components . . . 59

B.2 Baseline Multi-Objective MILP Program . . . 62

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List of Figures

1.1 Schematic of the compartment layout in forward and aft holds of the A321 baseline aircraft. Door locations indicate access routes to each compartment. . . 3 3.1 Coordinates of each compartment in relation to the reference

point of the baseline aircraft. . . 9 3.2 CG envelope with example ZFW and TOW balance states. A

balance state is valid if it is located within the appropriate boundary. . . 11 3.3 Left panel: Boxplot over the mass of different load items in-

cluded in the test data. Right panel: Histogram of the passen- ger seating base moment in the test data.. . . 28 4.1 Left panel: Relative frequency of each combination of selec-

ted compartments to load. Red bars are combinations with

>5% frequency. Right panel: Relative frequency of compart- ment combinations for each load item type. . . 31 4.2 Left panel: Histogram of ground stability slack. Right panel:

Histogram of final CG values for ZFW and TOW. . . 31 4.3 Left panel: Solution time boxplot for each solver. Right panel:

Time series of solution time for each test case iteration. . . 32 4.4 Left panel: Relative frequency of each combination of selec-

ted compartments to load. Red bars are combinations with

>5% frequency. Right panel: Relative frequency of compart- ment combinations for each load item type. . . 33 4.5 Left panel: Histogram of ground stability slack. Right panel:

Histogram of final CG values for ZFW and TOW. . . 33 4.6 Left panel: Solution time boxplot for each solver. Right panel:

Time series of solution time for each test case iteration. . . 34

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LIST OF FIGURES ix

4.7 Left panel: Relative frequency of each combination of selec- ted compartments to load. Red bars are combinations with

>5% frequency. Right panel: Relative frequency of compart- ment combinations for each load item type. . . 35 4.8 Left panel: Histogram of ground stability slack. Right panel:

Histogram of final CG values for ZFW and TOW. . . 35 4.9 Left panel: Solution time boxplot for each solver. Right panel:

Time series of solution time for each test case iteration. . . 36 4.10 Left panel: Relative frequency of each combination of selec-

ted compartments to load. Red bars are combinations with

>5% frequency. Right panel: Relative frequency of compart- ment combinations for each load item type. . . 37 4.11 Left panel: Histogram of ground stability slack. Right panel:

Histogram of final CG values for ZFW and TOW. . . 37 4.12 Left panel: Solution time boxplot for each solver. Right panel:

Time series of solution time for each test case iteration. . . 38 A.1 Left panel: Stable balance state on ground with CG forward

of pivot point. Right panel: Unstable state, CG aft of pivot point. . . 51 A.2 Fuel curve for the baseline aircraft. Fuel states with a pos-

itive moment contribution are of interest. Others are set to zero. . . 52 A.3 Exact non-linear CG along with linear bounding plane . . . . 56

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List of Tables

1 Table of abbreviations used in the report, in order of appearance xi 3.1 Tests flights per case. . . 26 3.2 Test model parameters . . . 28 5.1 Centre of gravity statistics in %MAC for ZFW in blue and

TOW inred. . . 40 5.2 Infeasible tests per case. . . 42 A.1 Standard passenger masses including handluggage. . . 53 A.2 Baseline aircraft parameters used for evaluation of ground

stability margin. . . 57 A.3 Required baseline parameter changes to achieve an actual

balance state with either 2%MAC or no margin on ground stability. . . 58 B.1 Moment arms from reference, volume and maximum allowed

weight for each compartment. . . 60 B.2 Cumulative weight limits for the baseline aircraft. . . 60 B.3 Balance envelope data for the baseline aircraft. Maximum

ZFW is colouredblueand maximum TOW isred. . . 61

x

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LIST OF TABLES xi

List of Abbreviations

Table 1: Table of abbreviations used in the report, in order of appearance Abbreviation Definition

CG Centre of Gravity

ULD Unit Load Devices

ACT Additional Centre Fuel Tanks

ZFW Zero-Fuel Weight

TOW Takeoff Weight

LAW Landing Weight

MILP Mixed-Integer Linear Programming

LMILP Lexicographic Mixed-Integer Linear Programming

CPT Compartment

CUM Cumulative

SEQ Sequence

FWD Forward

LMC Last Minute Changes

BY Normal Bags

BP Priority Bags

BT Transfer Bags

C Cargo

M Mail

CPU Central Processing Unit

RAM Random Access Memory

NLP Nonlinear Program

LP Linear Program

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Chapter 1 Introduction

Aircraft weight and balance is a critical safety concern in all aviation sec- tors, whether it be for small propeller aircraft or military and commercial jets.

The requirement is to ensure that prior to flight, the aircraft’s total weight and centre of gravity (CG) location are within the limits defined by its man- ufacturer. If violated, the aircraft may not fly. To satisfy this requirement, a decision has to be made on where to load all items to be carried on board.

Based on this decision, instructions on how to load need to be released and upon loading completion, a final check of the limits has to be made. In com- mercial aviation, this process is termed as load control. The complication in the process is the decision phase, termed as load planning, which can both be highly variable in the input parameters, the aircraft type and in the require- ments and preferences demanded by the operator on the output.

The most common method to complete this process is for a human agent to interface with a computerised calculation tool, where the agent manually al- locates a position to each item inside the aircraft. The software then returns the results for the aircraft balance state and whether all limits are satisfied.

The decision process is therefore entirely under the agent’s control and the software mainly aids with the required calculations. Certain industry soft- ware of this kind include some form of an algorithm to carry out this decision process when triggered by the agent. As most implementations are propri- etary and embedded in commercial software, details on their function is not readily available. Even so, the process is far from being automated in the industry and will in the vast majority of cases be completed manually.

Fuel efficiency is a key focus point for operators and one that specific CG locations can increase. The main feature of three major algorithm implement-

1

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2 CHAPTER 1. INTRODUCTION

ations [1–3] is optimal load planning with respect to fuel efficiency by proxy of the CG. However, simply optimising only with respect to the CG location may return a load plan that is only partially useful. It may take too much time to complete, require too many resources, be unsafe for ground operations or stability, risk mishandling of baggage and negatively affect its delivery time to customers, as well as potentially compromising handling operations on other flights awaiting transfer load from the flight in question. Over the allowed CG range of a commercial aircraft, the maximum attainable improvement in fuel burn is on the order of 1-1.5% [2]. Altering the CG slightly away from a spe- cific optimal target will thus only impact fuel efficiency in a marginal manner, while it may improve efficiency in other areas. This justifies trading-off CG optimality in a controlled manner to obtain a more overall efficient load plan, while still being fuel efficient.

In an online article from Lufthansa Systems [4] on the drawbacks of these kinds of algorithms, this specific point is discussed. From the perspective of the aircraft operator, i.e. an airline, the CG optimality appears as the only real criteria for aircraft loading, as it may have a directly attached monetary value.

How the aircraft is loaded can also impact other stakeholders involved in the process, i.e. those that actually have to load the aircraft, which in many cases are contractors of the airline. They may need to both allocate resources and execute loadings in an efficient manner, which is determined by the required loading actions. Lastly, the airline can also be negatively impacted as certain loadings may increase the risk of delays and dissatisfied customers due to mishandled or lost baggage and longer delivery times. These concerns may not necessarily have a direct cost impact, but certainly an indirect one, and thus they need to be considered in the process.

The handling of these multiple requirements of a good load plan, in terms of safety, the centre of gravity and its efficiency is the purpose of this thesis.

An optimisation model is developed where both constraints and objectives are designed to produce results that satisfy requirements from the various stake- holders involved.

1.1 Problem Description and Scope

Loading a commercial aircraft can be done in two distinct ways. The first method is to load each load item, e.g. bags, cargo boxes or mail pouches, individually piece-by-piece into the lower cargo compartments of an aircraft.

The other method is to first consolidate all items into containers, known as

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CHAPTER 1. INTRODUCTION 3

unit load devices (ULDs), of standard dimensions and then load and lock the containers into gridded positions inside the cargo compartments. Aircraft loaded by the first method are said to be bulk loaded and others are container- ised. Smaller commercial aircraft, e.g. the Boeing 737 and the Airbus A320 family, are mainly bulk loaded while larger aircraft, e.g. Boeing 747, 767, 777, 787 and Airbus A330, A340, A350 and A380, are containerised.

The problem investigated in this project will only focus on bulk loaded pas- senger aircraft, as they have features that are not present when dealing with containerised aircraft. These features are primarily volume uncertainty of load items as they come in different shapes and sizes, and the control of mixing between different items in each compartment. Containerised aircraft handle these features implicitly as the containers have a standard size and are pre- loaded.

The load planning problem of interest will use the Airbus A321 aircraft as a baseline platform. This aircraft is bulk loaded and has a very similar lower compartment layout to other similarly sized aircraft. This type is unique in one key aspect in that some versions may be fitted with additional centre fuel tanks (ACT) for extended range, which are located in one of the lower com- partments. For safety reasons, these fuel tanks impose specific restrictions on the aircraft loading which are of interest to model. Figure 1.1 shows the compartment layout of the A321.

Figure 1.1: Schematic of the compartment layout in forward and aft holds of the A321 baseline aircraft. Door locations indicate access routes to each compartment.

The items to be carried are transit, priority and normal bags along with cargo and mail. Quantities of each item are assumed known for each flight case

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4 CHAPTER 1. INTRODUCTION

along with passenger seating, so that only the loading of the lower compart- ments needs to be determined. Carriage of special items or dangerous goods cargo is not considered, as such items are normally only present in relatively small quantities compared to the aforementioned items, and require specific industry regulations to be implemented.

Load planning is to be performed in such a way to satisfy the following cri- teria:

• Satisfy all aircraft manufacturer limitations on weight and centre of gravity.

• Ensure ground stability of the aircraft during loading and offloading of both cargo compartments and passengers in cabin, i.e. to avoid tail- tipping.

• A satisfactory centre of gravity relative to a target range is achieved whenever possible.

• Place load items such that priority items can be offloaded first and all bags always offloaded before cargo and mail.

• Avoid mixing of incompatible load items, e.g. keep transit bags separ- ate from other bags.

• Use only a minimum number of compartments on board the aircraft to reduce the need to relocate equipment and personnel.

• Be capable of handling a variation in the input data with respect to aircraft limitations, e.g. last minute additions or reduction in load items after the load planning phase was completed.

Aircraft limitations are the only fundamental requirements that need to be satisfied for the aircraft to be airworthy. If all other criteria are not satis- fied the aircraft may still depart. Adherence to the other criteria is however preferred to increase operational efficiency and reduce the risk of errors. Al- though ground stability appears to be a critical criteria in the process it is generally not defined in industry standards or by manufacturers. It is thus up to the operator to have policies in place to satisfy this criteria. Gener- ally this is done by keeping the CG as far forward as possible by loading the forward compartments first and then loading the aft. For offloading the sequence is reversed. Some operators will define limits on the maximum al- lowed weight difference between forward and aft compartments to reduce the risk of tail tipping. Ground stability will be modelled by assuming these in- dustry procedures. Other criteria on the loading/offloading sequence, mixing

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CHAPTER 1. INTRODUCTION 5

and compartment usage are intended to ensure efficient operations prior to departure and after arrival so that load items can be processed depending on their inherent urgency and required service level.

A final requirement is for the load plan to be able to withstand perturbations in the input data so that all aircraft limits are still satisfied and no load items have to be moved to accommodate additions. This arises due to the fact that load planning in practice is generally performed before all input data is fully known. Passenger check-in normally closes between 45-60 minutes before departure but for large airport operations, the load plan needs to be available well before that to allocate the required resources. Operators will generally place a strict time deadline for cargo and mail acceptance to ensure that such information is readily available in time for load planning while using booked passenger numbers and average values for expected bags.

1.2 Project Goal

The goal of the project is to investigate whether a multi-objective optimisa- tion approach can be used to solve the bulk aircraft load planning problem according to the stated criteria in the problem description. A model will be developed and subsequently tested on a range of flight cases to evaluate the output, the robustness of the model and the sources of any infeasibilities en- countered.

1.3 Thesis Structure

A review of the available research literature related to the aircraft load plan- ning problem is presented in chapter 2. The methodology of the solution approach along with the underlying theory of aircraft weight and balance are then presented in chapter 3. Details of the model and test data are also presented in chapter 3. The thesis is concluded with a discussion of the ob- tained results and final conclusions on the entire project in chapters 5 and 6 respectively. Supplementary content specific to modelling of the ground sta- bility constraint and the baseline optimisation problem used in the thesis are presented in appendices A and B respectively.

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Chapter 2

Literature Review

Research into aircraft load planning began in the late 1970s with the introduc- tion of computerised weight and balance calculation tools. Prior to that, load plans were produced by hand using tables and graphs to determine CG loca- tions. This process is slow and tiresome for load controllers and limits their throughput. With increase in air travel, a significant improvement in resource productivity was achieved with computerisation [5].

Computerisation is however limited in the assistance it provides to the user.

Generally, these only implement basic aircraft weight and CG limitations. No aid is provided to ensure ground stability or adherence to operator loading procedures and trust is placed on the knowledge of the load controller. As a result, the agent may only be allowed to create load plans for a limited num- ber of operators and the operator may need to implement a quality monitoring program to ensure that procedures are being followed. These limitations re- quire investment in additional resources to both service and monitor the load planning process, which may be omitted if the required procedures are imple- mented in the computerised tools being used.

The bulk of academic research into optimal aircraft load planning has focused on large cargo aircraft where flexibility in load distribution is larger than for passenger aircraft. Larsen and Mikkelsen (1978) [6] consider a heuristic ap- proach to load cargo pallets on board a Boeing 747 configured to carry both passengers and cargo on its main deck. Their algorithm aims to minimise al- terations to loading at intermediate stops and to achieve a CG in the centre of the allowed range. This maximises the efficiency of the loading with respect to handling operations.

6

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CHAPTER 2. LITERATURE REVIEW 7

Mongeu and Bes (2003) [7], Verstichel et al. (2011, 2014) [8, 9] consider loading of containers and pallets on board larger passenger and cargo aircraft, where the objective is to maximise the value of the items loaded and aim for a target CG value. The items are selected from a pool of candidates which may not all fit on board. A mixed-integer linear program is used to model and solve the problem. No consideration on loading efficiency or ground stability is included in the model.

Limbourg et al. (2011) [10] consider optimal loading of pallets and containers on board a large cargo aircraft, minimising deviation from a target CG point with a mixed-integer program. Two approaches are presented; one where CG is optimised directly and another that minimises the moment of inertia of loaded items around the target point with a constraint on allowed CG values.

The latter is observed to be a better conditioned objective that implements the inside-out preference for aircraft loading. The model considers aircraft structural limits and CG envelope, but omits ground stability considerations.

Further aspects to handling efficiency are not addressed.

Limited research was discovered for smaller bulk loaded aircraft. Li et al.

(2010) [11] provide the most relevant work to the scope of this project. A rule- based approach, using pre-defined decision logic, is developed for a North- American carrier. Aircraft limitations are implemented into this decision pro- cess and evaluated with the preferred loading for each flight case. If satisfied, the loading is approved but if not, an alternative solution has to be available and then tested. This approach performs well for the specified carrier but may not apply to others which have different procedures. The whole decision logic thus has to be created again for each case.

Although some proprietary and commercially intended work may exist, the lack of public research into the load planning process of bulk aircraft makes the chosen topic an area of interest. The published models for larger cargo aircraft share some features with bulk aircraft, which can be implemented in the optimisation approach. However, they have not included considerations for efficient handling of load items or ground stability. Furthermore, the trade- off handling between the multiple objectives has neither been investigated nor published.

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Chapter 3

Methodology

This chapter details the methods used to determine the balance state of the baseline aircraft given a specific loading arrangement and provides a descrip- tion of the proposed optimisation model. A general model will be presented with all required formulae to build a computer program for testing.

3.1 Theoretical Framework

A review of the weight and balance calculation method and the fundamental ideas of the chosen multi-objective solution method are presented in this sub- section.

3.1.1 Aircraft Weight and Balance

The weight and balance calculations used to determine the balance state of a commercial aircraft involve a known layout representation of the aircraft along with a coordinate system defined by the manufacturer. Each passenger seat and compartment has a known moment arm around a given reference point which allows for a computation of total moment for a chosen loading arrangement. With the empty weight and CG state of the aircraft known, the final state can be determined. Bulk loaded aircraft generally only require the longitudinal CG to be determined as the mass can primarily be distributed along that direction. Larger aircraft may also require the lateral CG to be evaluated, but this is not required in the project scope.

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CHAPTER 3. METHODOLOGY 9

As stated in the problem description, it is assumed that the balance state of the aircraft loaded only with passengers in the cabin is known, and is therefore the base balance state. Knowing the balance state implies knowing both the total weight and total moment around the given reference point, which allows the CG to be determined.

The centre of gravity of the baseline aircraft needs to be determined for two different weights, the zero-fuel weight (ZFW) and the takeoff weight (TOW).

Other aircraft may also require the landing weight (LAW) CG to be determ- ined, but as it is determined in a similar manner, its inclusion can be easily implemented if required. The ZFW CG is determined from the balance state of the fully loaded aircraft without usable fuel. As most fuel is carried in the wings, this value can be taken as representing the fuselage mass distribution.

Limits on its value are required to reduce stresses on the airframe and ensure dynamic stability of the structure in flight. The TOW CG adds the weight and moment impact of the fuel carried at takeoff. Limitations are primarily required to ensure sufficient performance and safe flight handling character- istics. As fuel is burned, the CG will change during flight. The limitations of the baseline aircraft do not require intermediate in-flight balance states to be evaluated, as they are automatically satisfied by the ZFW and TOW limits.

This may not be a general case for all aircraft, but is observed for those of similar size.

Figure 3.1 shows the baseline aircraft placed in an arbitrary coordinate system with the location of the moment reference point preffrom the origin. Let hkbe the coordinate location of the k-th compartment from the origin. The moment arm of each compartment around the reference thus becomes

ak= hk− pref. (3.1)

Figure 3.1: Coordinates of each compartment in relation to the reference point of the baseline aircraft.

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10 CHAPTER 3. METHODOLOGY

Let W0 be the base mass of the aircraft in kg, loaded only with passengers, and M0 be the associated moment in kg·m around the reference point. If wk is the mass loaded in the k-th compartment, the total ZFW mass and moment become

WZFW= W0+

5

k=1

wk, (3.2)

MZFW= M0+

5 k=1

wkak= M0+

5 k=1

wk(hk− pref). (3.3) To obtain the appropriate values for TOW, the total fuel mass and moment are added to the above equations. The ZFW CG value can thus be determined as

aCG, ZFW= hCG, ZFW− pref= MZFW

WZFW, (3.4)

with the TOW CG obtained in a similar fashion. The value aCG, ZFW repres- ents the distance and direction of the CG relative to the reference point. The industry standard is to use a non-dimensional representation of the CG loc- ation relative to and normalised with the aircraft’s mean aerodynamic chord (MAC), which has fixed forward and aft stations along the fuselage and thus a fixed length. Let hMAC, fwdand hMAC, aft be the forward and aft stations re- spectively. The change to MAC representation of ZFW CG is obtained with the linear mapping

MACZFW=hCG, ZFW− hMAC, fwd

hMAC, aft− hMAC, fwd. (3.5) The MAC representation of CG will be used in the remainder of the report.

Limits on CG are given by the CG envelope, which shows limits on both total mass and moment that together determine the balance state according to (3.4). Figure 3.2 shows the envelope for the baseline aircraft with an example of valid ZFW and TOW balance states. For a given mass, the envelope gives upper and lower limits on the allowed moment values.

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CHAPTER 3. METHODOLOGY 11

Figure 3.2: CG envelope with example ZFW and TOW balance states. A balance state is valid if it is located within the appropriate boundary.

3.1.2 Lexicographic Mixed Integer Linear Programming (LMILP)

The method proposed to solve the multi-objective optimisation problem de- veloped in the project is the lexicographic method [12]. This method in- volves solving a sequence of mixed-integer linear programs (MILP), which are ordered in a lexicographic manner according to importance. A general overview of the method is presented in this subsection.

Assume an ordered set of linear objective functions fi(x) with i = 1, 2, . . . , n where x is a vector of both real and integer decision variables. The ordering of the objectives represents the preference of that function to be optimised.

This means that f1(x) is more important than f2(x), f2(x) is more important than f3(x) and so on. Applicable to all objectives is a set of linear equality and inequality constraints in the decision variables given in matrix form by

Ax≤ b, (3.6)

Cx= d. (3.7)

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12 CHAPTER 3. METHODOLOGY

Assume the vector partition x = [y, z], where y contains real valued variables and z takes integer values. For i = 1, define the initial MILP as

(P1) min f1(x) s.t. Ax≤ b,

Cx= d, x≥ 0, y∈ R, z ∈ Z,

(3.8)

which can be solved with any MILP solver. Take f1 to be the optimal value obtained from (P1). Let δ1 be an arbitrary positive parameter that represents an allowed deviation in the value of f1(x) relative to f1. Define a bounding constraint on the objective as

f1(x) ≤ f1+ δ1. (3.9)

If δ1 = 0, the bound effectively reduces to an equality constraint as the ob- jective was minimised. For i = 2, define (P2) as the next MILP

(P2) min f2(x) s.t. Ax≤ b,

Cx= d,

f1(x) ≤ f1+ δ1, x≥ 0,

y∈ R, z ∈ Z,

(3.10)

where the allowed absolute deviation in f1(x) has been added as a constraint and thus possibly restricts the feasible region for f2(x) optimality as allowed by the lexicographic ordering of the objectives. After obtaining the optimal value f2from (P2), the same procedure is applied to construct (P3) with devi- ating value δ2for f2(x), while keeping the constraint on f1(x) from (P2), and so forth.

A sequence of n MILP problems are solved and the vector xn with optimal value fnis taken as the solution to the multi-objective problem. As each prob- lem involves minimisation of the objective, only an upper bound on the op- timal value is required. For maximisation, the deviations would create lower bounds.

The lexicographic method is related to goal programming, which was first developed in the 1950s and 1960s. In goal programming, it is assumed that target values are known for each objective, which are added as equality con- straints with upper and lower slack variables representing deviations from the target. A single objective is used where a weighted sum of slacks between

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CHAPTER 3. METHODOLOGY 13

all objectives is minimised, and the weights can describe priority ordering as in the lexicographic method [12]. Variations of goal programming include a sequential solution approach, similar to the lexicographic method, where the ordering describes a clear priority ordering of the objectives. The main difference between the two approaches is that the lexicographic method does not require the target objective values to be known prior to constructing the problem.

A common drawback with goal programming formulations appears when ob- jectives are incommensurable, i.e. where magnitudes and units of the values does not allow a concise arithmetic comparison, thus requiring normalisation with the target objective or any other value. The structure of the lexicographic method does not suffer from this problem as it does not involve comparison between objectives. Another strength of the method is its relative simplicity, with it being easily implemented, altered and understood within any stand- ard optimisation framework and requires only a single parameter choice per objective function.

As aircraft load planning may involve objectives of different types, with dif- ferent units and magnitudes, which can be ordered by importance or prefer- ence, it is taken as a candidate for the lexicographic method.

3.2 Model

The components of the multi-objective MILP model proposed for the load planning problem are presented in this section. First, the index sets used are defined, followed by variables, constraints and finally the objectives. A general overview of each component is presented along with required formu- lations, which will subsequently be applied to the baseline aircraft for testing.

The overall model for the baseline aircraft is given in appendix B. Notation used in §3.1.2 does not apply to the model.

Three types of constraints are defined; modelling, aircraft limits and opera- tional. Modelling constraints are general limits imposed on the defined vari- ables due to modelling considerations, with aircraft limits and operational constraints set by the type of aircraft used and operator preferences. Each group is presented separately.

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14 CHAPTER 3. METHODOLOGY

3.2.1 Index Sets

Index sets for aircraft holds, compartments and load items are;

Holds = {H1, H2, H3, . . . , HnH}, (3.11) Compartments = {CPT1, CPT2, CPT3, . . . , CPTncpt}, (3.12) Bags = {B1, B2, B3, . . . , BnB}, (3.13) Cargo = {C1, C2, C3, . . . , CnC}, (3.14) Mail = {M1, M2, M3, . . . , MnM}, (3.15) where nH, ncpt, nB, nC and nM are the total number of elements in each set.

Elements in the {Bags} set are the different classes of bags, while elements in {Cargo} and {Mail} can be taken as individual shipments or different group- ings of shipments from a load planning perspective, where each has known total pieces and weight. All compartments have a parent hold. Thus, nHnon- empty and non-overlapping subsets of compartments belonging to each hold are defined as

SubCptj= {k | compartment k belongs to hold j}, (3.16) with

SubCptH1∪ · · · ∪ SubCptH

nH = Compartments, (3.17)

SubCptHj∩ SubCptH

j∗ = ∅, j, j∈ Holds and j 6= j. (3.18) For simplicity, let the union {Load} = {Bags} ∪ {Cargo} ∪ {Mail} be the set of all load items.

3.2.2 Variables

Two types of variables are used; decision and quantity variables. Decision variables are binary and take the value 1 if a hold or compartment is used and if a load item is loaded in a specific compartment. Quantity variables state how much of a given load item is loaded in a compartment where it has been decided to load. For items that may be split, the total quantity can be divided between holds or compartments but non-splittable items are constrained to a single location.

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CHAPTER 3. METHODOLOGY 15

Decision Variables

Let xjbe a binary variable indicating if hold j ∈ {Holds} is used.

Let ykbe a binary variable indicating if compartment k ∈ {Compartments} is used.

Let zik be a binary variable indicating if load item i ∈ {Load} is loaded in compartment k ∈ {Compartments}.

Quantity Variables

Load items can be loaded by pieces or weight. In general, bags are loaded in pieces and cargo and mail by weight. The quantity variables need to reflect the chosen method. Total pieces and weight of each item is known and constrains the values on these variables.

Let lik be non-negative integer variables describing the quantity of load item i∈ {Load} loaded in compartment k ∈ {Compartments}.

Let Qibe the known total quantity to be loaded of item i, in pieces or weight as desired.

3.2.3 Model Constraints

Decision and quantity variables are naturally coupled. If a specific hold or compartment is not used, the corresponding quantity variables for all items need to be zero. Conversely, if a hold or compartment is used then quantity variables must be non-zero.

Hold Selection

Based on xj value, the corresponding constraint on compartment usage for {SubCpt}Hj is

xj

k∈SubCptH j

yk≤ xj|SubCptHj|, j∈ Holds, (3.19)

where | · | is the set cardinality operator. If hold j is not used, then xj= 0 and all ykare constrained to be zero, indicating that no compartment belonging to hold j can be used. If hold j is used with xj= 1, then at least one compartment needs to be used and at most the total number of compartments in the hold.

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16 CHAPTER 3. METHODOLOGY

Compartment Selection

Based on ykvalue, constraints on load item decision for compartment k are yk

i∈Load

zik≤ yk|Load|, k∈ Compartments. (3.20) If yk= 0 then compartment k is not used and all associated load item decision variables are constrained to be zero. If not, at least one item needs to be loaded and at most the total number of items to load.

Compartment Loading

Decision variables zikdirectly control the limits on quantity variables for each load item as per

zik≤ lik≤ zikQi, i∈ Load, k ∈ Compartments. (3.21) If zik= 1, then at least 1 kg or 1 piece of item i needs to be loaded in com- partment k and at most the total quantity Qi.

It is assumed that all items need to be loaded, which requires the following equality constraints

k∈Compartments

lik= Qi, i∈ Load. (3.22)

Splittable Items

Let {NoSplit} be a subset of {Load} for all non-splittable items, that is NoSplit = {i | if item i ∈ Load is non-splittable}. (3.23) Splittability is a binary feature, thus an item can either be split or not. No limit to the number of splits is set. The splitting constraint is

k∈Compartments

zik≤ 1, i∈ NoSplit. (3.24)

3.2.4 Aircraft Limits Constraints

Aircraft limits are either for structural or balance purposes. Structural limits are related to compartment volume and maximum allowed weight. Balance is restricted by the centre of gravity envelope.

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CHAPTER 3. METHODOLOGY 17

Compartment Volume

Let vi denote the volume per piece or per weight, i.e. density, of load item i. Let pik be a packing factor of load item i in compartment k. The packing factor is a tunable parameter to change the effective volume of an item based on its loading location.

Let Vk be the available volume in compartment k, which yields the volume constraints

i∈Load

pikvilik≤ Vk, k∈ Compartments. (3.25)

Maximum Compartment Weight

Let wi denote the weight per piece of load item i. The maximum allowed weight in compartment k is Wk, thus yielding

i∈Load

wilik≤ Wk, k∈ Compartments. (3.26)

If load item i is loaded by weight, then wi= 1.

Cumulative Weight Limit

Subsets of compartments may be subject to cumulative weight restrictions.

These subsets need not be the same subsets of compartments as those of com- partments belonging to each hold.

Let {CumCpt}rbe the r-th such subset, with r = 1, 2, . . . , ncumand ncumas the total number of sets. For each r, the cumulative weight limit is Wcum,r which yields

k∈CumCpt

r

i∈Load

wilik

!

≤ Wcum,r, r= 1, 2, . . . , ncum. (3.27)

Centre of Gravity Envelope

Centre of gravity is limited for three aircraft weights, zero-fuel (ZFW), land- ing (LAW) and takeoff weight (TOW), denoted by WZFW, WLAW and WTOW

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18 CHAPTER 3. METHODOLOGY

respectively. Each weight must not exceed its maximum allowed value W× according to

WZFW≤ WZFW, (3.28)

WTOW≤ WTOW, (3.29)

WLAW≤ WLAW. (3.30)

Balance limits are represented by lower and upper bounds on total moment around the reference point for each weight, denoted by MZFW, MLAW and MTOWrespectively.

Let the base moment for each weight be M0,ZFW, M0,LAWand M0,TOW and the moment arm of compartment k from the reference be ak. Total moments are then obtained as

MZFW = M0,ZFW+

k∈Compartments

ak

i∈Load

wilik

!

, (3.31)

MLAW = M0,LAW+

k∈Compartments

ak

i∈Load

wilik

!

, (3.32)

MTOW= M0,TOW+

k∈Compartments

ak

i∈Load

wilik

!

. (3.33)

Total moment for each weight is limited from above and below by M× and M× respectively for each weight. Let ε×, ε×≥ 0 be known fixed additional margins for the upper and lower moment limits respectively. The bounds then become

MZFW+ εZFW≤ MZFW≤ MZFW− εZFW, (3.34) MLAW+ εLAW≤ MLAW≤ MLAW− εLAW, (3.35) MTOW+ εTOW≤ MTOW≤ MTOW− εTOW. (3.36)

Additional Centre Fuel Tanks

Installation of additional centre fuel tanks (ACT) into a hold may introduce restrictions on allowed loading to protect the tanks. No general form of these exist, thus only the specific version applicable to the baseline aircraft will be formulated.

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CHAPTER 3. METHODOLOGY 19

Assume that installed ACTs affect nACTsubsets of compartments. Let {ACT}s be the s-th such subset

ACTs= {k | if compartment k is affected by the s-th ACT limit}. (3.37) The elements in {ACT}s are assumed ordered by the sequence in which the ACT restricts the compartments.

The constraint type modelled takes the following form:

1. Compartment loading shall start immediately at the ACT and then by each subsequent compartment. Assume that the ordering of each {ACT}s reflects this.

2. A non-initial compartment cannot be loaded unless the previous in the sequence is sufficiently full with respect to volume. The sufficient limit is a specific percentage of the total available volume.

3. If a compartment is not sufficiently full, any subsequent compartments not affected by item 2 may instead be subjected to alternative weight restrictions.

Let yACT,k be a binary variable for each k ∈ {ACT}s with ordinality ko¯ <

|ACTs|, indicating whether compartment k is at least qks% volumetrically full.

The ordinality of compartment k only needs to be less than the cardinality of the ordered set ACTs as the loading state of the last compartment has no subsequent compartment to restrict. The volume constraints (3.25) can thus be adjusted for these compartments as

qksVkyACT,k

i∈Load

pikvilik≤ qksVk 1 − yACT,k +VkyACT,k, k∈ ACTs,

with ko¯ < |ACTs|, for s = 1, 2, . . . , nACT,

(3.38) where o¯ is the ordinality indicator. If yACT,k = 0 the constraint becomes the normal volume constraint (3.25) with the upper bound changed to qks% of the total available volume Vk, as the compartment should be less than qks% full.

If yACT,k= 1, the volume loaded in the compartment must be between qsk% and 100% of the total available volume Vk.

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20 CHAPTER 3. METHODOLOGY

Compartment usage variables yk in the sequence up to a cut-off point ¯ko

¯

s are affected by yACT,k as

yk≤ yACT,k, k, k∈ ACTs, (3.39) with 1 < ko¯ ≤ ¯ko

¯

s, and ko

¯ = ko

¯ − 1, for s = 1, 2, . . . , nACT.

This constraint implements the sequential loading of the impacted compart- ments such that for a given compartment k, it may not be used unless the previous compartment kin the set {ACT}s is at least qsk% full, as defined by the constraints (3.38). The cut-off point ¯ko

¯

s is defined as a control para- meter, since the entire set may not have to be affected by this restriction.

Let the alternative weight restrictions on compartments behind an insuffi- ciently full compartment be Wk. Their application is controlled by yACT,k and modifies the maximum compartment weight constraints (3.26) as

i∈Load

wilik≤ WkyACT,k+Wk 1 − yACT,k , k, k∈ ACTs, (3.40) with ko¯ > 1,

and ko

¯ = ko

¯ − 1, for s = 1, 2, . . . , nACT. If the preceding compartment k is not sufficiently full, then the alternative restriction is applied to the compartment k.

Application of ACT Constraints to Baseline Aircraft

As the ACT constraints described above are in general form, their applica- tion to the baseline aircraft is given here for completeness and clarity. As shown in figure 1.1, the aircraft has an ACT installed in compartment 3, reducing the available volume for loading. This installation affects the en- tire aft hold, i.e. compartments 3, 4 and 5. A single ordered set {ACT} = {CPT3, CPT4, CPT5} is defined according to the formulation.

The binary variables yACT,kare defined for compartments 3 and 4 only, as the last compartment does not have any subsequent compartments to restrict. An 80% compartment volume loading is considered as sufficiently full for these two compartments, thus qCPT3 = qCPT4= 0.8. Constraints (3.38) and (3.39) are constructed accordingly.

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CHAPTER 3. METHODOLOGY 21

The loading sequence constraints (3.39) only apply to compartments 3 and 4. Therefore, the cut-off index is CPT4. Compartment 4 can thus only be loaded if compartment 3 is at least 80% full. Compartment 5 is allowed to be loaded if compartment 4 is not 80% full, but its loading is instead restricted to a maximum of 250 kg. If compartment 4 is more than 80% full, the normal weight restriction applies. Let the associated alternative weight restriction for compartment 5 be WCPT

5 = 250 and construct constraint (3.40) accordingly.

3.2.5 Operational Constraints

The aircraft’s ground stability along with restrictions on mixing and sequen- cing of load items in compartments are types of operator specified constraints.

Although ground stability is inherently an aircraft limitation, the fact that it is generally not provided by the manufacturer but rather defined by the operator places it in this category.

Ground Stability

Developing a constraint to ensure ground stability of the aircraft during load- ing and unloading requires more elaborate modelling than other constraints.

The details of the formulation are presented in appendix A, with only the result stated here.

The constraint limits a hypothetical worst-case scenario of the balance state during ground operations, i.e. where all passengers are considered of the heaviest type and are seated aft of the reference point. Variables for com- partment loading are restricted so the hypothetical CG is kept forward of a critical maximum aft tail-tipping limit. Should this hypothetical case avoid tail-tipping, then all others will have a more forward CG and thus be further from the aft limit.

Let Wstaband Mstabbe the ground stability mass and moment of the aircraft for a given flight case, as defined in appendix A. Let aCG, pivotbe the CG location corresponding with the aircraft pivot point. The ground stability constraint is Mstab≤ aCG, pivotWstab. (3.41)

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22 CHAPTER 3. METHODOLOGY

Load Item Mixing

Let dk,i,i be a binary parameter indicating whether load item i and i can be mixed in compartment k. The constraints are

zik+ zik≤ 1, k∈ Compartments, (3.42) i, i∈ Load,

if dk,i,i = 0.

Thus, mixing of items i and iin compartment k is prohibited if dk,i,i = 0.

Load Sequencing

Let the t-th ordered set of compartments k in descending order of unloading be {Seq}t. The number of such sets is nseq. Let si be a non-negative integer indicating the priority for offload of load item i. Multiple items may have the same priority. The constraints are

zik+ zik≤ 1, i, i∈ Load, (3.43) k, k∈ Seqt,

t= 1, 2, . . . , nseq, if si> si, and ko

¯ > ko

¯ .

The constraints prohibit loading of item i into compartment k in {Seq}t if another item iof lower priority has been loaded in another compartment k, which is unloaded before compartment k.

Priority Based Hold Selection

This constraint is defined due to the assumed offloading sequence used to ensure ground stability. Its purpose is to avoid load distributions where items of higher priority are loaded in the forward compartments, when items of lower priority are loaded in the aft. Such distributions will require lower priority items to be offloaded before higher priority ones.

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CHAPTER 3. METHODOLOGY 23

Let {FWD} and {AFT} be disjoint subsets of {Compartments} containing the forward and aft compartments respectively. Using si from §3.2.5, the constraint becomes

zik+ zik ≤ 1, i, i∈ Load, (3.44) k∈ FWD,

k∈ AFT, if si> si.

3.2.6 Objectives

Three objective functions are defined for the multi-objective problem. Based on the criteria listed in the problem description, these are:

1. Achieve a centre of gravity value as close to a target value, or be within a defined target range whenever possible.

2. Use a minimum number of compartments on board the aircraft to avoid the need to relocate personnel and equipment during the loading and offload processes.

3. Leave volume available at door locations in case last minute additional load items need to be added after the planning process.

The lexicographic order of each objective is according to the enumeration.

Other considerations in the problem description have been defined as hard constraints.

Target Centre of Gravity

The centre of gravity objective is defined for a general aircraft weight, where one is chosen as the objective for a specific flight. Either ZFW or TOW CG are most often used for the objective, depending on the operator. Assume that a target value bhCG is given. For a specific flight case, this value can be transformed with the chosen weight to a target moment value

Mb×=

bhCG− pref

W×=abCGW×. (3.45) Let εCG, εCG≥ 0 be upper and lower error variables forMb×, which define the range constraint

−εCG≤ M×−Mb×≤ εCG. (3.46)

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24 CHAPTER 3. METHODOLOGY

This constraint is added as a default constraint in the model. The objective function is the error from the target moment, given by

fCG= εCG+ εCG. (3.47)

Number of Compartments Used

The compartment use variables yk for k ∈ {Compartments} are used to define this objective function as the sum

fCPT=

k∈Compartments

yk. (3.48)

Last Minute Changes

The purpose of this objective is to obtain a load distribution that moves as much load away from door locations to free up space for last minute additions, which can be loaded in those positions without having to move previously loaded items.

For simplicity, the non-overlapping ordered loading sequence sets {Seq}t with t = 1, 2, . . . , nseqare assumed to also describe compartment sequences re- lative to each door position, where the first compartment in each set is closest to the door, followed by the second compartment and so on. The objective is then to move all items loaded in each set as far aft as possible, without bleeding any load items between sets.

Assume that a load distribution is given, where the total quantity of load item iover the t-th sequence set is Qit. The no-bleed constraint can thus be defined as

k∈Seq

t

i∈Loads

lik= Qit, t = 1, 2, . . . , nseq, (3.49) which only allows redistribution between compartments within each sequence.

The objective function is a weighted sum over the mass loaded in each com- partment k ∈ Seqt, weighed by each compartment’s ordinal number ko

¯

t within the set, given by

fLMC= −

nseq

t=1

k∈Seqt

i∈Load

wilikko

¯

t . (3.50)

As the multi-objective model is defined for minimisation of the objectives, but the weighted sum described above needs to be maximised, the negative sign is applied to convert it into a minimisation.

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CHAPTER 3. METHODOLOGY 25

3.3 Solution Process

The multi-objective optimisation problem is solved by using the lexicographic method, described in §3.1.2. The method involves n = 4 steps to obtain a solution. Each step is described below:

1. Minimise the target centre of gravity error using the objective fCGand variables εCG and εCG from §3.2.6. Let the optimal value be fCG and optimal variables be εCGand εCG.

2. Fix the variables εCGand εCGindividually based on εCGand εCGas εCG= εCG+ δCG, (3.51) εCG= εCG+ δCG, (3.52) where δCG and δCG are the allowed deviations. They are defined as fractions of the slack in the envelope constraints for the chosen target CG mass in §3.2.4 as obtained with fCG .

Minimise the number of compartments used with the objective fCPT from §3.2.6. Let the optimal value obtained in this iteration be fCPT . 3. Constrain the number of compartments used to the value obtained in

step 2 with

fCPT= fCPT , (3.53)

which allows all solutions that use the same minimum number of com- partments.

As it is possible to obtain multiple solutions in this step for any flight case, the target CG error minimisation from step 2 is repeated in this step to choose the particular solution with the smallest CG error. Re- move constraints (3.52, 3.51), and optimise for minimum CG error. Let the new optimal CG objective value be fCG∗∗ with variables ε∗∗CGand ε∗∗CG. 4. Using ε∗∗CGand ε∗∗CG, add the constraints (3.51) and (3.52) again accord-

ing to the new optimal CG error.

Apply the constraints from §3.2.6 and minimise the new objective fLMC. Let the optimal objective be fLMC .

The solution vector from step 4 is the solution to the multi-objective problem.

References

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