K.E. STECKE
Using Mathematics to Solve Some Problems in Industry
Using Mathematics to Solve Some Problems in
Industry
Kathryn E. Stecke
University of Texas at Dallas
School of Management
Richardson, TX 75083-0688
kstecke@utdallas.edu
Abstract
This paper describes how some industrial problems can be solved using relatively simple mathematical models.
Models can be used also to provide insight into how industrial system components interact. This paper is useful
for master's level students in that it describes what a model is as well as the modeling function. Master's students
in engineering and MBAs and Ph.D. students who are interested in solving real industry problems should find
this overview of models useful. Different types of industrial problems are discussed. Finally, some models that
address industrial problems, such as inventory models, linear programming, network flow, decision analysis,
queueing models, and simulation are examined. Future industrial modeling situations are described.
Editor's note: This is a pdf copy of an html document which resides at http://ite.pubs.informs.org/Vo5No2/
Stecke/ (Volume 5, Number 2, January 2005)
areas of mathematics as well as the variety of industrial
problems to which mathematics has come to aid.
1. Introduction
Mathematics has been called the language of science.
Mathematics is used to solve many real-world problems in industry, the physical sciences, life sciences,
economics, social and human sciences, engineering,
and technology, for example. Mathematics was used
to build many of the ancient wonders of the world,
such as the pyramids of Egypt, the Great Wall of China, the hanging gardens of Babylon, the Taj Mahal of
Agra, and the like.
What does it mean to model a particular system? First
one needs to abstract the essence of a situation and
then represent it mathematically. This representation
is then mathematically manipulated to provide some
useful information. Finally, the knowledge gained
from this information should be translated into an action for the system, situation, or problem at hand. A
nice overview of modeling as well as developing
modeling skills is provided in Powell (2001).
Early modern industrial engineering modeling tools
were developed by Taylor, the Gilbreths, and Gantt.
They proposed procedures to improve operations and
check the progress of multiple work activities. The
early project management techniques included project
evaluation and review technique and the critical path
method. These are still used today to help manage
large projects.
2. Types of Models
Some mathematical models are used to generate possible candidate decisions or solutions to problems.
Other models can be used to evaluate particular, possible sets of decisions (see Suri, 1985).
3. Generative Models
Early mathematics (computations, statistics, and accounting) has been applied to operations problems,
in administration and in managing technical activities,
by public administrators, engineers, and managers.
The focus of this contribution is on the use of mathematics to solve industrial problems. We discuss many
INFORMS Transactions on Education 5:2(1-8)
Mathematical techniques such as linear, integer, dynamic, and nonlinear programming generate solutions
to particular problems. Other generative models include differential equations, network flow models,
1
© INFORMS ISSN: 1532-0545
K.E. STECKE
Using Mathematics to Solve Some Problems in Industry
decision analysis, number theory, tabu search, genetic
algorithms, fluid dynamics, and game theory. Generative models help to resolve complex situations and
can provide candidate solutions. Assumptions always
have to be made that simplify the problem, in order
to get a solution. Assumptions can make the original
problems unrealistic. Often the simplifying assumptions may have no bad effect on the applicability of a
found solution. Modeling skills help a person to develop a model that can really be used by industry.
7. Industrial Planning Problems
Many types of planning problems are solved using
mathematics. Aggregate planning problems involve
making decisions on workforce and capacity over a
long period of time, say over a year's time period. See
Nahmias (2005). These decisions are made assuming
a forecast of demand, and provide constraints on the
actual day-to-day operations. Many such real, large,
aggregate production planning problems are solved
using linear programming.
4. Evaluative Models
In automated manufacturing, a variety of planning
problems need to be solved before actual production
can begin. In particular, a set of part types has to be
selected to be machined over some upcoming period
of time. (Integer programming can be used to select a
candidate set of part types. Simulation or queueing
models or Petri nets can be used to evaluate candidate
solutions (sets of part types) according to the appropriate measures of performance).
Other types of mathematical models can be used to
evaluate solutions to various industrial problems according to different, relevant performance measures.
Some evaluative models include queueing models and
queueing network theory, Petri nets, decision models,
data envelope analysis, simulation, and perturbation
analysis. Many of these are discussed in subsequent
sections.
The cutting tools required for each operation of each
selected part type have to be allocated to some machine
or machines, again before production can begin.
Nonlinear integer programming has been used to solve
such problems. Again, simulation or queueing theory
or Petri nets have been used to evaluate the goodness
of the solutions.
5. Models and Solution Algorithms
It is important to distinguish models and solution algorithms. A model describes a situation. Models are
a (usually mathematical) representation of a problem.
A solution algorithm finds one (or more) solutions to
the situation or problem. There could be a variety of
algorithms or methods to solve the problem.
8. Industrial Scheduling Problems
For example, as we will see soon, the total cost equation is a model for an inventory problem. We could
"solve" this model either using calculus, or by setting
up a spreadsheet, or perhaps just by graphing it in
Excel.
There are many types of scheduling problems. The
solutions to aggregate capacity planning problems
provide constraints to detailed scheduling problems.
For example, the number of workers of different types
and different skill levels may have been decided at an
aggregate level. These workers need to be allocated to
(usually) 8-hour shifts over 5-day work weeks, perhaps
over several shifts/day. Also these workers may need
to be allocated to workstations (machines or equipment).
6. Industrial Design Problems
These and other mathematical models have been used
to solve a variety of industrial problems. Some industrial design problems include the design of aircraft
and automobiles (using computational fluid dynamics,
for example) and the design of a manufacturing system
(using closed queueing networks or simulation, for
example).
The aggregate production planning problems are actually quite simple ... because so much information
has been aggregated! Linear programming then suffices to provide a good solution.
However, disaggregating and detailed minute-byminute scheduling is a very difficult problem, because
INFORMS Transactions on Education 5:2(1-8)
2
© INFORMS ISSN: 1532-0545
K.E. STECKE
Using Mathematics to Solve Some Problems in Industry
there are usually so many (too many) scheduling
possibilities to consider. Unless a workforce scheduling
problem is very small, it is very difficult to optimize
to find a very good solution (workforce schedule).
Often, heuristics, or rules of thumb, are used to try to
find an acceptable feasible schedule.
When a company invests in inventory, it costs money.
The inventory carrying cost for each item is iC, where
C is the purchase (or variable) cost of each item and i
is the inventory carrying charge as a percentage. Costs
that go into i include the opportunity cost of capital,
warehouse costs, insurance on the inventory, finance
costs, waste, and obsolescence.
The difficulty of the scheduling problems increases
when the goods that need to be manufactured are
considered. The production of every product type (i.e.,
an automobile) requires many components (e.g., to
assemble). The production of every component may
require many operations of different types (i.e., milling,
drilling, forming, and tapping). Each operation requires some time to process it, some workstation to
process it on, and somebody to perform the operation.
There is a partial precedence among the operations:
some operations have to be finished before others can
begin; for some operations, it doesn't matter which are
performed first.
Also, there are costs involved every time an order is
placed for Q units of the product. These costs include
telephone calls, faxes, stamps, i.e., all costs involved
in placing an order. The portion of time that the clerks
spend in placing the orders needs to be considered.
Other costs per order may include expediting costs
per order or setup costs per batch.
Let S be the ordering or setup cost per order. If Q items
are ordered every time an order is placed, D/Q is the
number of orders per year and SD/Q is the total annual
ordering cost. With Q/2 = average inventory, for any
order quantity Q, the total annual cost is:
Various raw materials have to be on hand to begin
some operations. These are inventory problems, discussed in the next section. The problem difficulties
increase when the due dates for the different products
(or customer orders) need to be considered. Many
different mathematical models have been used to solve
both the variety of industrial scheduling problems and
to evaluate the goodness of proposed solutions. Of
course a wide variety of performance measures exist.
Each is appropriate under different circumstances and
for different types of scheduling problems.
TC(Q) = SD/Q + iCQ/2+CD.
Graphically, we can see this relationship in Figure 1.
If the total annual cost includes the sum of the annual
ordering cost and annual inventory carrying cost, we
can see the following tradeoff. If Q is large, the annual
inventory carrying cost is high, but the annual ordering
cost is low. Conversely, if Q is small, the annual inventory carrying cost is low, but the annual ordering cost
is high. Calculus can be used to determine the optimal
Q, the amount to order (every time an order is placed)
to minimize the sum of the total, relevant, annual costs.
By differentiating TC(Q) with respect to Q, setting the
equation equal to zero, and solving for Q, we get the
EOQ, which is
9. Inventory Problems
Another large category of problems, for which many
mathematical models have been proposed to address
various types of issues, is inventory. Inventories cost
money and are subject to obsolescence. Much is written
about just in time inventory systems; often no inventory is a goal. Yet some inventory is necessary to allow
production operations to run smoothly.
Examining this relationship more closely, we can see
some problems one might run into when trying to
operate just in time (JIT). JIT aims to reduce inventory
and inventory costs by ordering exactly the right
amount when it is required, usually by ordering small
amounts. We see from the above discussion that ordering a small amount each time does decrease inventory
carrying costs. But then orders are placed more often,
so total ordering costs increase. JIT only makes economic sense when the cost per order is small.
To present some basic inventory issues, let's start with
the most simple inventory problem. Suppose that there
is a constant demand for an expensive product ($C)
that is part of an assembly. Given an annual demand
D for this product, calculus can be used to find the
optimal amounts of inventory, Q, to order every time
an order is placed. We can see the cost tradeoffs in the
following.
INFORMS Transactions on Education 5:2(1-8)
3
© INFORMS ISSN: 1532-0545
K.E. STECKE
Using Mathematics to Solve Some Problems in Industry
models, while considering deteriorating efficiencies
of existing models. The model uses dynamic programming. Input parameters include material use, energy,
emission factors, and fuel economy over a 36-year time
horizon.
Baker (2000) describes five nice linear programming
problems and solutions. Insights into the problems
are provided by analyzing the sensitivity of the optimal solutions. Other linear programming problems
and their applications are provided in Hillier and
Lieberman (2005) and Nahmias (2005).
Figure 1: Total Cost as a Function of Order Quantity.
11. Network Flow Models
This very basic, simple, inventory model can be generalized in many ways. Including quantity discounts for
items or transportation costs can increase the complexity of the equation describing the total annual cost of
dealing with inventory. Another cost component (annual purchase cost) now depends on Q. But quantifying these cost components has many useful purposes.
It is important for a firm to know how much it is
spending on inventory; the firm has to set reasonable,
competitive prices for its items. More important, by
knowing these costs, the firm can make intelligent
decisions concerning its inventory policies to decrease
the total annual cost of dealing with inventories.
1974 (and 1975) Nobel Prize winners Tjalling Koopmans (and L.V. Kantorovich) were the first to propose
network flow models. In the early days of World War
II, Koopmans modeled the problem of moving people,
supplies, and equipment from various U.S. bases to
foreign bases. The goal was to optimize one or more
of: minimizing total transportation cost, minimizing
total transit time, and/or maximizing defensive effectiveness. A few years earlier, the Russian mathematician and economist Kantorovich used network flow
models to address some important problems in the
Soviet economy. In particular, he investigated the
problems of allocating production levels among factories and distributing the resulting products among the
markets. They seem to be the first to structure and
analyze complex decision problems as network flow
problems.
Other complications of such simple inventory models
include consideration of the lead time required between when an order is placed and when the shipment
arrives. Even more complicating is the situation when
the exact lead time is uncertain. Many additional cost
components often should be considered, such as
transportation or warehouse costs.
Glover, Klingman, and Phillips (1992) provide many
examples of network flow model applications. Some
applications that they mention include electrical circuit
board design, telecommunications, water management,
the design of transportation systems, metalworking,
chemical processing, aircraft design, fluid dynamic
analysis, computer job processing, production, marketing, distribution, financial planning, project selection,
facility location, and accounting. They also provide
other, non-industrial applications of network flow
analysis, in the arts, sociology, archaeology, and more.
Considering inventory requirements at every stage of
a complex supply chain is important. Inventory problems occur when considering an appropriate number
of spares to have on hand to be able to cope with disruptions of various kinds.
10. Mathematical Programming
Mathematical programming has been used often to
provide solutions to many industrial problems. Linear
and nonlinear programming problems have been described in earlier sections. A good example can be
found in Kim et al. (2003). They develop a life cycle
optimization model to determine optimal vehicle lifetimes, accounting for technology improvement of new
INFORMS Transactions on Education 5:2(1-8)
Still other applications of network flow models discussed in Glover et al. (1992) include airline revenue
management, employee shift scheduling, and best use
of energy resources. They also provide references for
actual applications.
4
© INFORMS ISSN: 1532-0545
K.E. STECKE
Using Mathematics to Solve Some Problems in Industry
For example, a large automobile manufacturer uses
many computer hours every week running a network
model that determines how many cars of each model
type to produce at various plants and after, how many
automobiles of each type to ship to various cities. See
Figures 2 and 3.
such models to determine optimal flight training
schedules for pilots. Timing and classroom availabilities are considered. An automobile manufacturer uses
a network flow model to determine optimal batch sizes
for various plastic components required for car assembly.
One major oil company uses a discrete network model
and software to decide on the optimal timing of oil
use and distribution from its leased and owned wells
over several years. Another oil company uses an optimization-based network decision support system for
supply, distribution, and marketing planning.
A major US chemical company uses network models
to determine customer zones to be served from its
warehouses, and for different alternative commodity
and freight mode combinations. The Tennessee Valley
Authority uses a discrete network model and solution
procedure to determine the minimum cost refueling
schedule for nuclear reactors. Texas uses a network
model to choose the locations of out-of-state tax audit
offices. These collect sales taxes and other taxes owed
to Texas. The model also assigns states and auditors
to these offices.
Figure 2: Map of Automobile Plants and Major Markets.
Chemical products companies use a network flow
model to integrate production, inventory, and distribution operations. They also use it to determine plant
location and sizing.
These are but some of the industrial applications
modeled and solved using networks. Glover et al.
(1992) provide references that detail most of these applications. Methods to solve these problems, both
discrete and continuous, are also given.
12. Decision Analysis
There are many useful techniques to structure problems that have multiple objectives. These include
multi-attribute utility theory and the analytic hierarchy
process. These are useful for when a group needs to
make a decision. AHP uses pairwise comparisons of
alternatives to help arrive at a "good or best choice".
Figure 3: A Network Model Representation of Automobile
Plants and Major Markets.
Figure 4 shows a flowchart of a decision analysis process. Variations of this chart can be conceptualized.
An international pharmaceutical company uses network flow models and software to decide on which
drugs and medications and how much of each that it
should produce. A lumber company uses a network
flow model to schedule logs to plants, and logs to
products, and logs to markets. The U.S. Air Force uses
INFORMS Transactions on Education 5:2(1-8)
A survey by Keefer et al. (2004) provides references
to many applications of decision analysis methods.
Some of these applications include real options, purchasing equipment, new product strategies, telecommunication applications, and others.
5
© INFORMS ISSN: 1532-0545
K.E. STECKE
Using Mathematics to Solve Some Problems in Industry
Figure 4: A Decision Analysis Process Flowchart.
Keeney and Raiffa (1976) provide many examples of
decision-making problems under uncertainty. There
are often preferences for possible consequences or
outcomes. Some of their examples are:
•
Safety of landing aircraft;
•
Strategic and operational policy concerning frozen
blood;
•
Sewage sludge disposal in the metropolitan Boston
area;
•
Electrical power versus air quality;
•
Airport location;
•
Selecting a job or profession;
•
Heroin addiction treatment;
•
Transporting hazardous substances;
•
Medical diagnostics and treatment;
•
Development of water quality indices;
•
Business problems (profit versus ethics);
•
Airport development for Mexico City.
•
Hospital blood bank inventory control;
•
Air pollution control;
•
Fire department operations;
•
Siting and licensing of nuclear power facilities;
INFORMS Transactions on Education 5:2(1-8)
Details as well as complete references concerning these
examples can be found in Keeney and Raiffa (1976).
6
© INFORMS ISSN: 1532-0545
K.E. STECKE
Using Mathematics to Solve Some Problems in Industry
13. Queueing Models
15. A Future Industrial Application
Queueing models have been used to investigate industrial problems for many years. In the 1940s, queueing
models were used to solve a variety of machine interference problems, i.e., how many repairpersons to assign to maintain a system, or how many telephone
operators to handle traffic calls. (See Stecke, 1992 and
Palm, 1943)
A future application falls in the realm of homeland
security. Many supply chains are particularly vulnerable to disruptions because of their design characteristics and operating philosophies. Disruption effects on
direct targets could be substantial. Long lasting and
rippling effects could be felt throughout multiple
business sectors because of the increase in business
interrelations. A current complex JIT environment
creates supply chains that integrate many private and
public entities with unclear contingency plans and
roles in a disaster. JIT also provides insufficient buffers
to absorb unusual system disturbances in supply networks. Since most companies' operations are not flexible enough for essential quick responses, disruptions
can create bullwhip and queueing effects. The amplification of demand variability from a bullwhip effect
can cause increasing negative economic effects further
upstream and downstream.
Queueing models are used to analyze tradeoffs concerning the number of servers versus the waiting time
of customers. Clearly, if the number of servers is high,
the cost of the servers is high, but the waiting time
(cost of customer idle time) is low. Notice that this is
the same kind of tradeoff discussed in the basic inventory order quantity decision discussed earlier.
Some kinds of queueing problems involve determining
the appropriate number of service facilities to cover
expected demand, as well as determining the efficiency
of servers and the number of servers of different types
at the service facilities. (See Hillier and Lieberman,
2005) These are design problems and decisions are
made in order to provide an appropriate level of service. Suri (1998) suggests using queueing theory to
provide quick solutions to industrial problems.
There is a need to understand these systems to probe
for weaknesses, predict outcomes, and test policies.
The scale and complexities involved pose significant
problems. Schmitt and Stecke (2003) have applied
novel methods of simulating and modeling to synthesize the huge, dynamic mass of information that flows
in supply networks. They are working with Sandia
National Laboratories in the development of an "agentbased" model of supply networks of 14,000 manufacturing firms in the Pacific Northwest to assess the
economic impact of various disruptions in critical infrastructure. The overall mathematical modeling effort
should contribute to the U.S.'s effort and ability to
prepare for possible attacks on critical infrastructure
such as electric power, telecommunications, and
transportation, and improve the effectiveness and efficiency of responses should such attacks occur.
14. Simulation
Simulation is a potentially very detailed modeling tool
that is widely used to evaluate the solutions to many
industrial problems. Many measures of the performance of a system are available. A great benefit of
simulation is that a model can often be very close to
reality. This can sometimes result in a complicated
simulation model that can take a long time to run.
Another drawback is that (without perturbations
analysis), a simulation can model only one system at
a time, with one set of parameters. It is difficult to
generalize the results from a simulation.
16. Acknowledgements
The author would like to thank Sanjay Kumar at the
University of Texas at Dallas for his help in locating
some material for this piece. The author would also
like to thank Erhan Erkut and the Associate Editor for
their excellent suggestions and comments on the earlier
submissions of this piece.
However, with the detailed modeling capability of
simulation, the results from the model of a particular
system can be quite accurate. Simulation has been a
widely-used, very useful modeling tool. Many examples, showcasing the modeling aspects of simulation,
can be found in Schriber (1991).
INFORMS Transactions on Education 5:2(1-8)
7
© INFORMS ISSN: 1532-0545
K.E. STECKE
Using Mathematics to Solve Some Problems in Industry
Suri, R. (1998), Quick Response Manufacturing, Productivity Press, Portland, OR.
References
Baker, K. (2000), "Gaining Insight in Linear Programming from Patterns in Optimal Solutions," INFORMS Transactions on Education, Vol. 1, No.
1,
http://ite.pubs.informs.org/Vol1No1/Baker/
Suri, R. (1985), "An Overview of Evaluative Models
for Flexible Manufacturing Systems," Annals
of Operations Research, Vol. 3, pp. 13-2.
Glover, F., Klingman, D., and N. Phillips,(1992), Network Models in Optimization and Their Applications in Practice, John Wiley & Sons, New York.
Hillier, F. and G. Lieberman, (2005), Introduction to
Operations Research, Eighth Edition, McGrawHill, New York.
Keefer, D.L., Kirkwood, C.W., and J.L. Corner (2004),
"Perspective on Decision Analysis Applications," Decision Analysis, Vol. 1, No. 1, pp. 4-22.
Keeney, R.L. and H. Raiffa, (1976), Decisions with
Multiple Objectives: Preferences and Value Tradeoffs, John Wiley & Sons, New York.
Kim, H.C., Keoleian, G.A., Grande, D.E., and J.C Bean
(2003), "Life Cycle Optimization of Automobile
Replacement: Model and Application," Environmental Science & Technology, Vol. 37, No. 23, pp.
5407-5413.
Nahmias, S. (2005), Production and Operations Analysis,
Fifth Edition, Irwin McGraw Hill, Homewood
IL.
Palm, D.C. (1943), "Intensitatschwankungen in Fernsprechverkehr, or Analysis of the Erlang Traffic
Formula for Busy-Signal Arrangements," Ericcson Technics, Vol. 6, pp. 39.
Powell, S.G. (2001), "Teaching Modeling in Management Science," INFORMS Transactions on Education, Vol. 1, No. 2,
http://ite.pubs.informs.org/Vol1No2/Powell/
Schriber, T.J. (1991), An Introduction to Simulation Using
GPSS/H, John Wiley & Sons, New York.
Schmitt, T.G. and K.E. Stecke (2003), "Simulating Disruptions in Critical Infrastructure within Electronics Company Supply Chains in the Northwestern United States," European Applied Business Conference, Venice, Italy.
Stecke, K.E. (1992), "Machine Interference: Assignment
of Machines to Operators," Handbook of Industrial Engineering, Second Edition, Gavriel Salvendy, Editor, John Wiley & Sons, New York.
INFORMS Transactions on Education 5:2(1-8)
8
© INFORMS ISSN: 1532-0545