Army logisticians often have to make quick decisions
about how to support the warfighter based on incomplete data
and illinformed assumptions. Making such decisions is a necessary
skill for logisticians, because an 80percent solution that
arrives on time is better than a 100percent solution that
is late. The ability to approximate a solution to a logistics
problem and then accomplish the mission is a valuable skill
for any logistician. However, when they have time to plan and
analyze courses of action, logisticians need to look for optimal
or improved solutions to logistics problems that consider all
functional constraints. This is important because the best
possible solutions save resources and time while maximizing
a unit’s ability to support.
Operations research (OR) is a branch of applied mathematics
that uses algorithms, simulation, modeling, queuing, and stochastic
methods to optimize or improve a realworld situation. OR was
developed by a group of British and American mathematicians
who were studying strategic logistics problems during World
War II. Since that war, this branch of mathematics has been
used in a variety of industrial and military applications.
For example, at New Mexico State University, an ongoing OR
project for the New Mexico Chile Task Force is optimizing a
supply chain network involved in the harvesting, transporting,
and processing of red chile peppers. In the project, data and
statistics fitting is used to assign probability distribution
functions to crop maturation, harvesting rates, transportation
rates, crop damage, and processing rates. After assigning valid
queuing rules to the system, the research team was able to
write a program using ProModel, a discreteevent simulation
modeler that accurately mirrored the actual conditions. Using
this program as a base, the team adapted the model to quantify
the value of lengthening the growing season, optimizing transportation,
and constructing storage based on the amount of crop loss.
The project’s value was in creating a model of a supply
chain that used accurate stochastic data to accurately describe
and then improve on a complicated logistics system. Although
writing a program is not a valid method of solving problems
in an Army unit, logisticians can use OR methods to deduce
and compare feasible solutions to common problems.
This article will describe a few available OR techniques. Using
the references listed below, those interested can examine methods
for improving systems within their work areas.
Analyzing a Network
A useful OR technique is finding optimal solutions to problems
involving start nodes, arcs, and destination nodes. A basic
problem in OR is the “transportation problem,” in
which there are known supply bases, known customer demands,
and known costs to each route from supply base to customer.
(The costs could be in time, risk, shipping costs, or something
else that is considered important.) The objective is to minimize
total cost while meeting all demands. The chart below
shows an example of a transportation network and a sample
transportation network spreadsheet with costs.

At the top
is an example of a transportation network, showing
how each supply base (SB) supplies the demand (D)
of each customer. At the bottom, a spreadsheet indicates
the cost in time of each supply base meeting the
demands of each customer. 

This problem can be solved using either a program
or a number of algorithms performed by hand. In this example,
all demands
equal the sum of all supplies in order to simplify the spreadsheet.
In more realistic examples, a dummy variable is added to
balance the equation in order to perform either an algorithm
or a linear
program. After using an algorithm and checking with linear
algebra, the optimal solution is shown in the chart below.

This chart
shows the best way for the two supply bases to use
their supplies to meet the demands of the three customers.
The analysis, conducted by using an algorithm and
checking with linear algebra, divided the base’s
supplies as shown to meet the demands. 

Networks, such as maximum flow networks (which are useful
in describing port activities), decision trees, lattices,
and
other deterministic flows, can be used in many ways to
solve logistics problems. For those interested in learning
more,
see the references listed below.
Programming
In OR, programming is used to quantify a problem involving
an objective function that is subject to one or more constraints
in the system. An objective function attempts to perform
actions that affect the output of a system, such as minimizing
shipping cost, maximizing throughput, or maximizing material
shipped to an area. Constraints are functions that place
limits on the range of the objective function. These can
include limitations on infrastructure capacity, warehouse
space, cost, trucks available, and integer, or nonnegativity,
limits.
The chart (below) outlines a scheduling problem that
illustrates the usefulness of programming. In this example,
a maintenance shop must complete four jobs in a week using
three mechanics who are given different times to complete
the jobs.

This chart
illustrates a hypothetical scheduling problem in
a maintenance shop. Each mechanic needs the hours
shown to complete the four jobs. However, each mechanic
can work only a
40hour week. 

For this example, a number of constraints must
be listed. First, each worker can only log a maximum of 40
hours during
the week. Second, each job must be completed at the end
of the week. Third, only one worker can be assigned to
a job. For this problem, the decision variables are labeled
as “Xij” (assigning “i” worker
to “j” task) and will have a value of either “1” or “0” (that
is, to either assign or not assign a worker to a task).
The cost attribute is labeled as “Bij” (the
time for “i worker to complete “j” task).
The resulting problem can be written as shown in the chart
below. It is solved by using an optimizing algorithm,
called the simplex method, that uses linear algebra in
order to solve series of linear equations. [In nonlinear
programs (those involving objective functions and constraints
with higherordered variables), other methods, such as
interior point methods, linear approximation, and Hessian/Duality
matrices, must be used.] In realworld problems
with multiple variables and constraints, it usually is
not feasible to do calculations by hand. For this type
of work, many computer programs like MATLAB and LINDO are
available. The key to solving these problems is to set
up the problem correctly in order to describe the situation
and the desired type of solution accurately.

The solution
to the problem of scheduling the work of the three
mechanics so they can complete all of their work
in 1 week, with each of them working a 40hour week,
requires the use of an optimizing algorithm, called
the simplex method, that uses linear algebra to solve
series of linear equations. The bottom three lines
incorporate the constraints on the schedule. 

Simulation
Simulation involves using a combination of deterministic and
probabilistic functions to model the problem and then predict
actual system improvements after changes. Because of the
large number of calculations involved and the need for multiple
runs, simulations are almost always run on a computer. A
number of excellent simulation programs are available, such
as ProModel or Arena, that use graphical interfaces to help
model an actual system. Simple models, however, can be run
from Microsoft Excel.
The basis for the simulation is generating random numbers and
assigning ranges of numbers to values that fit a probability
distribution function (otherwise known as Monte Carlo simulation).
Although this type of simulation has been used in many scientific
disciplines, it was first used to calculate the properties
of the neutron by Enrico Fermi and Stanislaw Ulam in 1930.
To use Monte Carlo simulation, you could generate a uniform
series of random numbers from zero to one (U(0,1)) and then
assign an equal range to each probability in the following
way—
Using
a computergenerated pseudorandom number generator (PRNG),
you then could model the real situation of a die roll. More
complicated simulations, for example, could model fuel usage
in a division based on past data and assumptions about the
current mission.
In a realworld logistics model, the simulation would look
simultaneously at a variety of distributions where the means,
variances, and distribution types are fit by examination
of past data while adding in deterministic data based on
known parameters of the future mission. By running the simulation
in multiple iterations, you would be able to get a stasis,
or expected solution, along with data such as network utilization,
efficiency, and expected system variance.
OR is directly related to logistics because it focuses
on optimizing realworld tasks, such as designing an efficient
supply chain. We highly recommend that anyone who is contemplating
an advanced civil schooling assignment look into industrial/OR
engineering as an option. Most programs require an engineering
or science background and coursework in multivariable calculus,
differential equation, and statistics. We also recommend
that anyone pursuing a master’s degree review discrete/vector
mathematics and linear algebra before starting.
For those either not interested in advanced civil schooling
or not in that period of their careers, we recommend any
of the reference books in the list on page 49. Operations
research is a continuously evolving discipline that is
directly related to the main objective of logisticians—efficiently
supporting the warfighter.
ALOG
Dr. Delia J. VallesRosales is an assistant professor
of industrial engineering at New Mexico State University.
She received her Ph.D. degree in industrial engineering
from New Mexico State University. Her research specialties
are microelectromechanical manufacturing and packaging
and discreteevent simulation. She can be contacted at
dvalles@ad.nmsu.edu.
Major Donovan O. Fuqua is assigned to the 595th Transportation
GroupProvisional (SDDC) in Kuwait. He completed an M.S.
degree in industrial engineering/operations research at
New Mexico State University while on an advanced civil
schooling assignment and currently is working on his doctorate
in engineering with a specialty in operations research
and supply chain management and a minor in mathematics.
He can be contacted at donovan.fuqua@us.army.mil.