# INTEGER PROGRAMMING RESEARCH PAPERS by gabyion

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```									                                Sarada Sundar

INTEGER PROGRAMMING RESEARCH PAPERS

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PAPER 1
Title: Applying Integer Linear Programming to the Fleet Assignment Problem
Author: Jeph Abara
Source: Interfaces, Volume 19, Number 4, July-August 1989 (pp. 20-28)

Airline optimization is an area of extensive research and highly complex problems can be
modeled and solved to optimality or near optimality using Operations research tools. This
paper will be especially interesting to Integer programming students as it describes an
integer linear programming model developed by American airlines for optimal
assignment of its aircraft fleets to flight schedules. This report hopes to give the reader an
idea about the real-world scenario that is modeled in this paper, discuss the formulation
of the integer-programming model and its efficacy, enumerate the benefits gained and
identify areas for further improvement.

Purpose of the Fleet Assignment Integer programming model:
By considering available resources like aircraft and gates, American airlines creates a
schedule with repeating patterns of flights within it, that comprises of over 2300 flights
per day to 150 cities using more than 500 jets. The objective of the fleet assignment
process is to assign as many flight segments as possible in a schedule to American
Airline’s ten fleet types, while optimizing a certain objective function (saving operating
costs or maximizing profit) and meeting operational constraints (restriction of certain
flights to operate within certain aircraft types, restriction on the number of aircraft that
can remain overnight at particular stations, limits on arrivals and departures at a station
during the day) The integer program formulation, on being given a schedule with the
departure and arrival times indicated, solves the fleet assignment problem by determining
which flights should be assigned to which aircraft types to optimize the objective
function.

Constraints and objective function:
There are five main groups of constraints, out of which four are intrinsic to the model and
the fifth one includes all user specified rules.
Flight coverage constraint: Flight-to-flight connections are referred to as ‘turns’. An
arriving flight can typically turn to more than one departing flight, but the latter’s
departing time should permit a minimum time of 40 minutes for the connection. Also, no
more than one of a flight’s possible turns can be active, to prevent a flight from being
counted twice.
Continuity of equipment constraint: This ensures that each flight begins and ends on the
same aircraft type, even when there is a turn (flight-to-flight connection) to ensure
network integrity.
Schedule balance by station and aircraft type constraint: This maintains conservation of
flow by introducing origination and termination shortage variables. Physical imbalance in
the network is avoided by ensuring that the sum of the sequence originations is equal to
the sum of the sequence terminations with the shortage variables on either side of the
equation.

Aircraft count constraint: It minimizes the number of aircraft used by ensuring that
aircrafts of all types should be exhausted first before any new aircrafts are added for
schedules that are very large.
Other optional constraints use additive flight-related variables to place limits on the
number of aircraft remaining overnight at an airport, utilization of the aircraft, slots or
daily service and system operating costs. ‘Difficult’ constraints that need to be imposed
include placing limits on the number of stations served, since allowing a new aircraft type
to remain over night at a station may incur incremental maintenance personnel and parts
inventory costs. Invalid turns can be eliminated either by explicitly eliminating them or
imposing a penalty on them in the objective function. The objective function maximizes
the benefit contributions of the flights less the costs of aircraft used, cost of imbalances
and cost of stations.

Formulation:
Xijk is a binary decision variable which indicates that flight leg i turns to flight leg j on
aircraft type k. Ysk is another binary decision variable that indicates service or no service
of aircraft type k at station s. The formulation is simple and the paper provides detailed
theoretical explanations for every constraint that has been modeled. The problem size can
get very large for even medium sized schedules. Run times range from under two minutes
(for a two aircraft problem) to over sixty minutes for a problem involving four aircraft
types, while using an IBM 3081 machine.

Special features of the model and implementation successes:
Biases ensure that the model assigns larger aircraft to high load flights. This is achieved
by setting penalties for assigning small aircrafts to legs with the highest percentage of
traffic. This feature of the model has saved American airlines millions of dollars. The
model described has been used by American airlines for one-time decisions involving
fleet planning, crew base planning and schedule development. This model has allowed
the development of a crew base planning system that analyses the cost of a large number
of crew base scenarios. The model has been successfully used to increase the average
daily utilization of the aircraft fleet. Operating costs have been reduced and operating
margins increased. This model is one of the key modules that are going to be
incorporated into the next generation scheduling system currently being developed by
American airlines. Once this is done, the model will move forward from being an ad hoc
decision making tool to being used on a daily basis to develop current and future
schedules.

Usefulness of the paper and model efficacy:
The paper is easy to understand, even without prior knowledge about airline flight
allocation issues. The advantage of the ip model in this paper is that it takes into account
a number of constraints that mirror fundamental real world issues that must be considered
while assigning fleets to schedules. This model can be further extended upon to include
issues like flight cancellations and delays. A number of advances have taken place in the
field of airline optimization since the paper’s publication (in 1989) and this paper can
serve as a stepping stone for understanding papers analyzing more complex optimization
problems.

PAPER 2
Title: Optimal planning in large multi-site production networks
Authors: Christian H. Timpe, Josef Kallrath
Source: European Journal of Operational Research, 126 (2000), pp 422-435

Motivation for understanding this paper:
Operations research techniques can be applied to solve supply chain and manufacturing
related problems. Integer programming provides valuable tools for this purpose and thus
motivated my search for literature on ip techniques for solving production-planning
problems, considering demand and other commercial considerations.

Purpose of the paper:
This paper describes a mixed integer-programming model that was successfully
implemented in BASF. The problem is about making a production plan for multi-purpose
plants with machines that can be operated in different modes, producing different
products. The basic model for this problem has been formulated in a previously published
paper and is further extended in this paper to model many commercial aspects. This
model falls under the category of small bucket problems with two setups (one before and
one after mode change) per period. Special features like batch (integer multiples of
smallest unit produced) or campaign (several batches following each other immediately)
production, multi-level production and raw material consumption issues can be modeled.
The model supports a number of objective functions like maximizing sales and
contribution margins and minimizing costs.

Problem scenario described:
The model considers a number of multi-purpose production plants located in different
countries. There are a number of sales points that are connected to specific production
sites. The sales points might also be interconnected. All of them have storage tanks and
container space for different products. In order for the model to follow real-world
scenarios as closely as possible, different time scales are used; for plants, a production
time scale is used and for sales points, a commercial time scale is used to account for less
exact market forecasting.

Formulation:
The goal is to create an optimal production plan for a certain planning horizon, whose
starting and ending points can be chosen by the user. The entire planning horizon is
divided into a number of production slices by using the function of length of the tth
commercial time period over the integral number of production slices embedded in that
time period. Using the earlier model developed, the following variables can be obtained –
the variables that give information about the number of days (can also be fractional) in
which the plant is in a certain mode during a certain production period and the production
variables. With an inequality constraint, these variables are connected with the help of a
production rates data variable, which also indicates whether a product (p) can be
produced in a certain mode (m) in a site (i). In case of campaigns, they can be numbered
successively and an additional index can be added to the production variable to indicate
the campaign in which the product is produced.

Constraints and objective function:
An equality constraint describes the usage of raw materials with a usability factor (which
equals 1 when all the raw material is converted into products with no loss). Transport
variables are defined which indicate the tons per production period of product (p) shipped
from – site to sales point or between sales points. These variables are semi-continuous, as
the amount transported has to be above a certain minimum amount. Sophisticated MIP
solvers have capabilities for definition of variables of this sort. A distribution equation
distributes products from a plant to a site. Material balance equations describe the second
type of transport.
For describing the demand data, an additional index c is used which allows the model to
classify products into different categories. This allows different prices to be defined for
different customers and exclude certain origins for certain products. A constraint is
defined for ensuring that no more is sold than the required demand. Practical issues, like
sticking on to a particular origin for a customer throughout the planning horizon are taken
into consideration by making use of binary variables. The model allows for external
purchase of products if the internal capacity is insufficient. For this, an additional plant is
described as the origin for all products purchased externally. If the problem is infeasible,
all the products can be purchased externally then. The model considers stocking products
in tanks and containers. Binary variables are defined for keeping track of which product
is in which tank. They are coupled with inventory variables to ensure that tanks contain
only one product at a time and to enforce capacity constraints too.
Material balance equations connect all the different parts of the model and the
commercial time periods. Bounds are enforced on raw material inventory levels to ensure
that they are above some safety stock. The objective function can be modeled to
maximize contribution margin by subtracting the sum of a number of costs including
variable production costs, model changing costs, transportation costs and inventory costs
from the yield computed on the basis of production and associated sales prices. The LP-
relaxation of the problem is computed by the primal-simplex method. Then, branch and
bound techniques are used for solving the problem.

With the help of this model, short-term demands for extra deliveries can be speedily
accepted or rejected, tank storage levels can be reduced and at the same time the need for
certain tank storage levels can be proven. The disadvantage of the model is that it allows
for only one mode change per production period. The model reflects real-world scenarios
with a sufficient degree of reality. The production plans obtained are not counter-
intuitive. The model is a good tool for production planning in the chemical process
industry, but can also be extended to other industrial applications. It provides optimal
solutions in some cases, and even when it doesn’t, it can at least provide safe bounds. The
model plans for production, not just considering a single plant as an isolated entity, but
considering a multi-plant, multi-site scenario and acknowledging it as a part of an entire
supply chain. Innovative constraints accounting for customer preferences and allowing
for outsourcing of products makes this paper very relevant to real-world business issues.
The paper successfully demonstrates the adaptability of integer programming methods to
solving highly complex practical problems.

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