Exact and Heuristic Methods for Solving Large-Scale Integer Programs
Jonathan F. Bard
Graduate Program in Operations Research
& Industrial Engineering
The University of Texas
Austin, Texas 78712-0292, USA
Syllabus
This 1-day seminar will be divided into 5 units. The first unit will discuss exact methods for
solving integer programs, concentrating on column generation. The goal will be to show how an
intelligent partitioning of the constraints can be exploited to provide improved bounds and more
rapid convergence of branch and bound algorithms. The next 3 units will focus on industrial
applications. The first of these will describe a pure implementation of column generation for the
problem of scheduling nurses over a 1-month planning horizon. The second is aimed at
designing driver work areas for express pickup and delivery firms, and the third discusses a
variety of heuristics for assigning tasks to workers at mail processing and distribution centers.
The final unit will provide some insight into writing and publishing articles in high quality
journals. In addition, necessary and sufficient conditions will be presented for succeeding as a
faculty member at a research university.
Unit 1: Column Generation
Over the last few years great strides have been made in finding optimal or near-optimal solutions
to large-scale mixed integer programming (MIP) problems. The most prominent techniques
include column generation, Lagrangian relaxation, polyhedral theory, and intelligent heuristics,
all coupled in some way with branch and bound. In fact, it is rare that any one technique can be
applied successfully to solve MIPs that arise in practice. What is needed is a strategy that
combines insights about a particular problem, with lower bounding procedures, limited
enumeration, and simple methods for quickly finding good feasible solutions.
In general, difficult problems should be tackled by decomposing them into easier problems,
called relaxations, to obtain bounds that approximate the optimal value. Conventional branch
and bound uses a linear programming relaxation of the constraint-based formulation. The
Lagrangian approach consists of removing some of the constraints and placing them in the
objective function as penalty terms. The polyhedral approach involves improving the LP
relaxation by adding inequalities to strengthen the formulation. In the column generation
approach, Dantzig-Wolfe decomposition is used to create what is called an extensive
formulation, which is equivalent to a set-covering-type problem. The majority of this seminar
will be devoted to describing how a MIP can be transformed into a column format and then
solved to optimality. Related issues include initialization, variable selection for branching,
bound computation, and stability.
1
Unit 2: Cyclic Midterm Nurse Scheduling
Personnel scheduling in the service industry presents challenges that are absent in most
manufacturing environments where fixed-length shifts are the norm. Organizations such as
hospitals, call centers, airlines, and retail outlets typically operate up to 24 hours a day, 7 days a
week, and face widely fluctuating demand during the week. The consequences of poor planning
are often evidenced by low job satisfaction and morale accompanied by high turnover rates and
increased recruitment and training costs. This talk presents an improved methodology to solve
the cyclic preference scheduling problem for hourly workers. The application to be discussed
involves the develop of 1-month rosters for nurses. The objective is to strike a balance between
satisfying individual preferences and minimizing personnel costs.
To address this problem, a new integer programming model is presented that combines the
elements of both cyclic and preference scheduling. To find solutions, a branch-and-price
algorithm is developed that makes use of several branching rules and an extremely effective
rounding heuristic. An unusual feature of the formulation is that the master problem contains
integer rather than binary variables. Computational results are reported for problem instances
with up to 200 nurses. Most were solved within 10 minutes and many within 3 minutes when a
double aggregation approach was applicable.
Unit 3: Constrained Clustering for Rationalizing Pickup and Delivery Operations
Express package carriers must periodically redesign the work areas assigned to drivers as
demand and customers change. For a fleet of homogeneous vehicles and a given set of pickup
and delivery points, the objective is to find the least number of work areas or clusters that satisfy
a variety of geometric and capacity constraints. Using rectangles as the basic shape, each cluster
must have an aspect ratio that falls within certain bounds, as well as meet load and time
requirements dictated by the capacity of a vehicle and the working hours in a day. The latter
requirement presents a unique hurdle because travel times are a function of the actual routes
followed by the drivers, and are not known, even in a probabilistic sense, until the clusters are
formed.
In this talk, it is shown how the problem can be modeled using a compact set-covering
formulation and how solutions can be obtained with an adaptive column generation heuristic.
Because it is not possible to efficiently represent all the constraints in algebraic form, thus
allowing a Dantzig-Wolfe decomposition, it was necessary to take a constructive approach. The
first step involved generating a subset of attractive clusters from seed customers scattered
throughout the service region and then iteratively pricing them out to obtain a relaxed solution to
the set-covering model. To find integer solutions, a three-phase variable fixing scheme was
designed with the aim of balancing solution quality with runtimes. Test results will be discussed
for six cities in the U.S. using data provided by FedEx.
Unit 4: Task Assignment Problem
Assigning tasks to workers during their daily shifts is a common problem in several sectors of
the service industry. For a homogeneous workforce, a given set of workstations, and a
corresponding demand for labor, the objective is to develop a disaggregated schedule for each
worker that minimizes the weighted sum of transitions between workstations. In the
2
formulation, each day is divided into 48 ½-hour time periods and a multi-commodity network is
constructed in which each worker corresponds to a unique commodity and each node represents
a workstation-time period combination.
Initial attempts to solve large instances with a commercial code indicated a need for a more
practical approach. This led to the development of a reduced network representation in which
idle periods are treated implicitly, and a sequential methodology in which the week is
decomposed into 7 daily subproblems and each solved in turn. A tabu search metaheuristic was
also developed in an effort to improve upon the solutions. Results are reported for instances
associated with running a U.S. Postal Service mail processing and distribution center.
Unit 5: Academic Publishing
Universities that view their mission primarily as one of conducting research rather than one of
teaching, expect their faculty to publish in the leading journals and to establish a prominent
reputation in their fields. As a corollary, the need to obtain grants to support research, graduate
students and professional activities is implicitly understood, especially in engineering schools.
For many young faculty, the demands of academia may appear to be overwhelming and almost
impossible to achieve within a 24-hour day. To be successful, those starting out must learn to
prioritize their work and budget their time. The purpose of this talk is to provide some insights
and suggestions for selecting research topics, preparing manuscripts for publication, balancing
competing career forces, and achieving recognition in one’s chosen profession.
3