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**Airline Schedule Optimization (Fleet Assignment I)**

Saba Neyshabouri

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**Agenda Airline scheduling process Fleet Assignment problem**

Time-Space network concept

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Airline Schedule Single most important indicator of airline’s business strategy. Markets to be served Level of service There are many restrictions that makes the planning very difficult: Gates and slots Operational restrictions Airport Restrictions Location of the crew and maintenance plans Airline schedule is what that defines their level of service, their market share, and their business strategy, because it is the most important product of an airline and it is the core service which all the other services are based on that. The problem and decision making is a complicated process because of numerous complicating factors such as : Labor, Operational restrictions Aircraft restrictions Aircraft requirements Airport restrictions Gate, slots, Uncertainty in demand

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**Airline’s Goals Airlines are operating in a competitive market.**

The ultimate goal of airlines is maximizing the profit. There can be some other goals that will lead to profit such as: Operational goals Marketing goals Strategic goals Airlines are trying to find the best (in terms of profit) schedules that are consistent with their other goals. Airlines are private owned companies, and similar to all the other private owned companies, the ultimate goal is maximizing the profit,. In order to maximize profit, airlines have different marketing , operational and long term strategies which can be considered as goals. Because they will generate profit.

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**Airlines and Decision making**

Decision making process in airline industry is a very complicated process due to: Numerous airport location with different restrictions Different aircraft types with different operational characteristics Crew scheduling and regulations Large number of O/D routes and markets Decision making for Airlines in United States is very difficult. Competitive market Enormous Air transportation network. Airports Hubs Flight legs Markets Rules and regulations Crew Aircraft operations

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**Complicating Factors in Decision making**

In modeling and solving optimization problems in airline industry, 2 major complicating factor are known: The huge size of the problem Inherent uncertainty of the system The most complicating factor that makes the decision making in airline industry extremely difficult is the inherent uncertainty of the system Demand is the most uncertain element which is the driver of all the operations of airlines. Aircraft operations and processes have stochastic elements. Maintenance Crew availability On the other side, (not even considering the stochastic property of the problem) the size of the problem and decision space is so big that makes it intractable for some problem structures.

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**Breaking Down the Problems**

In order to handle airline’s operational problems, it has been broken down to several hierarchical problems: The schedule design problem The fleet assignment problem The maintenance routing problem The crew scheduling problem In order for airlines to be able to tackle their problems, they have to break it down to smaller pieces, Note that after breaking down the airlines operational problem, each sub problem is still very difficult and there has been extensive research on each topic. For example look for the searching airline crew scheduling problem in Google scholars will bring 9,220 results which shows that this problem is not that easy to solve and extensive research has been done to make this problem tractable. Same search for fleet assignment will give 10,100 hits

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**Fleet Assignment Problem**

The objective: Finding a profit maximizing assignment of aircrafts to flight legs in airline’s network. Complicating factors: Satisfying passenger demand Fleet composition Fleet balance (flow balance) Other side constraints Other side constraints can be airport restrictions in accepting some aircrafts, maintenance planning for aircrafts and the operational limitations of an aircraft

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**The Schedule Design Problem**

The goal is to design the airline’s flights schedule specifically: Flight legs to be operated by airline Scheduled departure times Estimated scheduled arrivals Frequency plan and the days that on which flight leg is operated Before fleet assignment problem, airlines should develop their schedule

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**Sample Flight Schedule**

This example for flight schedule connects only 3 markets and has 10 flights. These are 10 flights for connecting 3 different markets

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Example Flight network Fleet composition

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Example Given this example the goal is to find a profit-maximizing assignment of fleet types to flight legs in a way such that: Not more than available number of aircrafts are used Balance of aircrafts at each location is maintained The objective function tries to maximize the profit therefore the profit of assigning a fleet type to a flight leg should be calculated:

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Profit Calculation After doing the calculation for each possible assignment, the resulting profit for each assignment of fleet type to flight leg is summarized in the following table:

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Greedy Solution Greedy methods: heuristic method to find a solution to a complicated problem which reduces the time of computation however it is not guaranteed to be optimal or even feasible. The main idea of a greedy algorithm is to be greedy in each step of decision making! Being greedy is like not considering long-term effects of decisions. Being greedy in some cases might not even provide any feasible solution.

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**Greedy Solution to Example**

Considering the most profit generating assignments, the greedy solution will be: This solution is not feasible! In our example, being greedy is choosing the aircraft that maximizes the profit for that flight It is infeasible because aircraft balance is not satisfied It is difficult to know how many aircrafts are needed this solution does not tell you how many aircrafts of each type is needed to operate the schedule

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**Greedy Solution to Example**

This solution is not feasible! The aircraft balance is not achieved. Using a network of distances (static network) makes it difficult to determine the number of necessary aircrafts to fly for each day of operations

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Time-Space Networks In many problems in optimization, time is playing an important role in the model. However having time as a changing parameter in the model, usually increases the complexity of the problem in hand. Example of the problems that deal with time related constraints: Job shop scheduling- Minimizing tardiness Vehicle routing problem with time windows Flow shop scheduling problems with job availability constraints

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Time-Space Network Decisions that are needed to be made at different times require adding variables that keeps track of time. Time is a continuous variable! Adding a continuous variable to an IP problem makes the problem even more complicated to solve. There has to be an smart way to deal with time in our models.

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**Time-Space Network Concept**

Graph G=(N,E) is made of set of nodes (N) and set edges (E) N: usually represents the locations E: usually represents the arcs (connections/roads) between two locations N={ORD,BOS,LGA} E={CL50x,CL55x,CL30x,CL33x}

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Time-Space Network As it can be seen in the graph, there is no indication of the times of flights: However in managing the flights, keeping track of time is important since one aircraft can fly multiple legs.

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**Sample Time-Space Network**

In general, in time-space networks, each node represents a location in a specific time (of the day/month/year). Arcs are moving between two locations considering the time it takes for that movement. ORD So if there exists an arc that goes from BOS at 8 to BOS at 9 it means that the aircraft has stayed at BOS for that period of time. As it can be seen this way of modeling will give more fidelity and transparency in the decision making process. LGA BOS 8:00 9:00 10:00 11:00 12:00 13:00

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**Time-Space Network In our example: Not all the arcs exists.**

The size of the network is much bigger than the static network. ORD LGA BOS Not all the arc exists, also there can be a preprocessing that if there is a node (location in a time) that no arc is attached to it, it can be removed. 8:00 9:00 10:00 11:00 12:00 13:00

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**Time-Space Networks: Pros & Cons**

Time-space networks are used so the optimization problem does not become a mixed-integer programming (MIP) which are generally more difficult to handle. Using time-space networks, may cause the problem to transform into one of the well-known network problems which can be handled efficiently. Using time space network will cause the size of the problem to grow very fast N= Number of locations * Number of time windows (or significant times for each node) E= Every possible movements between 2 locations throughout the day. Network optimization has been studied and developed extensively and there are family of these network problems that can handled very efficiently

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**Time-Space Network for our Example**

In our example: a time-space flight network is an expansion of the static flight network in which each node represents both a location and a point in time. In this network, two different arcs are possible: A flight arc: representing a flight leg with departure location and time represented by the arc’s origin node, and arrival location and arrival plus turn time represented by the arc’s destination node. A ground arc: representing aircraft on the ground during the period spanned by the times associated with the arc’s end nodes.

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**Time-Space Network for our Example**

Our static network will change to another network that will capture the temporal behavior of the system: Flight arc Ground arc

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**Optimal Fleet Assignment**

In our network, the optimal fleet assignment is shown on the following network (Flow Balance): For the floe balance constraint, it can be seen that aircrafts do not get stuck at some point in time, and model will move aircrafts from one location to the other locations or at least in moves them in time so we can keep track of time.

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**Optimal Fleet Assignment**

In our network, the optimal fleet assignment is shown on the following network (Same location for aircrafts requirement): It can be seen that in this model the fleet balance model is maintained: B737 goes form BOS to ORD while the other one goes from the ORD to BOS so at the end of the day one of each aircrafts are sitting at each place.

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