Introduction to Operations

Research
OPERATIONS RESEARCH – A QUANTITATIVE PERSPECTIVE TO DECISION-MAKING
• Managerial activities have become complex and it is necessary to make right decisions to avoid heavy losses.
Whether it is a manufacturing unit, or a service organization, the resources have to be utilized to its maximum in an
efficient manner. The future is clouded with uncertainty and fast changing, and decision-making – a crucial activity
– cannot be made on a trial-and-error basis or by using a thumb rule approach. In such situations, there is a greater
need for applying scientific methods to decision-making to increase the probability of coming up with good
decisions. Quantitative Technique is a scientific approach to managerial decision-making. The successful use of
Quantitative Technique for management would help the organization in solving complex problems on time, wit
greater accuracy and in the most economical way.
• Operations research approach helps in the comparison of all possible alternatives (courses of action or
acts) with respect to their potential outcomes and then sensitivity analysis of the solution to
changes or errors in numerical values. However, this approach (or technique) is an aid to the decision-
makers’s judgement not a substitute for it.
While attempting to solve a real-life problem, the decision-maker must examine the given problem from
both quantitative as well as qualitative perspective. For example, consider the problem of investments in
three alternatives: Stock Market, Real Estate and Bank Deposit. To arrive at any decision, the investor
needs to examine certain quantitative factors such as financial ratios from the balance sheets of
companies whose stocks are under consideration; real estate companies’ cash flows and rates of return
for investment in property; and how much investment will be worth in the future when deposited at a bank
at a given interest rate for a certain number of years? Also, certain qualitative factors, such as weather
conditions, state and central policies, new technology, political situation, etc.?
• The evaluation of each alternative can be extremely difficult or time consuming for two reasons: First, the
amount and complexity of information that must be processed, and second the availability of large number
of alternative solutions. For these reasons, decision-makers increasingly turn to quantitative factors and
HISTORICAL DEVELOPMENT

• During the early nineteen hundreds, Fredrick W. Taylor developed the scientific management principle which was
the base towards the study of managerial problems. Later, during World War II, many scientific and quantitative
techniques were developed to assist in military operations. As the new developments in these techniques were found
successful, they were later adopted by the industrial sector in managerial decision-making and resource allocation.
The usefulness of the Quantitative Technique was evidenced by a steep growth in the application of scientific
management in decision-making in various fields of engineering and management.
• At present, in any organization, whether a manufacturing concern or service industry, Quantitative Techniques and
analysis are used by managers in making decisions scientifically. Quantitative Techniques adopt a scientific
approach to decision-making. In this approach, past data is used in determining decisions that would prove most
valuable in the future. The use of past data in a systematic manner and constructing it into a suitable model for
future use comprises a major part of scientific management.
• Application of scientific management and Analysis is more appropriate when there is not much of variation in
problems due to external factors, and where input values are steady. In such cases, a model can be developed to suit
the problem which helps us to take decisions faster. In today's complex and competitive global marketplace, use of
Quantitative Techniques with support of qualitative factors is necessary. Quantitative Technique is the scientific way
to managerial decision-making, while emotion and guess work are not part of the scientific management approach.
This approach starts with data. Like raw material for a factory, this data is manipulated or processed into
information that is valuable to people making decision. This processing and manipulating of raw data into
meaningful information is the heart of scientific management analysis
METHODOLOGY OF QUANTITATIVE TECHNIQUES
Formulating the Problem

• As a first step, it is necessary to clearly understand the problem situations. It is important to know how it is
characterized and what is required to be determined. Firstly, the key decision and the objective of the problem
must be identified from the problem. Then, the number of decision variables and the relationship between
variables must be determined. The measurable guaranties that are represented through these variables are
notified. The practical limitations or constraints are also inferred from the problem.
Defining the Decision Variables and Constraints

• In a given problem situation, defining the key decision variables are important. Identifying these variables helps us
to develop the model. For example, consider a manufacturer who is manufacturing three products A, B and C using
two machines, I and II. Each unit of product A takes 2 minutes on machine I and 5 minutes on machine II. Product B
takes 1 minute on machine I and 3 minutes on machine II. Similarly, product C takes 4 minutes and 6 minutes on
machine I and machine II, respectively. The total available time on machine I and machine II are 100 hours and 120
hours, respectively. Each unit of A yields a profit of Rs. 3.00, B yields Rs. 4.00 and C yields Rs. 5.00. What should
be level of production of products A, B and C that should be manufactured by the company so as to maximize the
profit?
• The decision variables, objective and constraints are identified from the problem. The company is manufacturing
three products A, B and C. Let A be x1, B be x2 and C be x3. x1, x2 and x3 are the three decision variables in the
problem. The objective is to maximize the profits. Therefore, the problem is to maximize the profit, i.e., to know
how many units of x1, x2 and x3 are to be manufactured. There are two machines available, machine I and machine
II with total machine hours available as 100 hours and 120 hours. The machine hours are the resource constraints,
i.e., the machines cannot be used more than the given number of hours.
• To summarize,
• l Key decision : How many units of x1, x2 and x3 are to be manufactured
• l Decision variables : x1, x2 and x3
• l Objective : To maximize profit
• l Constraint : Machine hours
Developing a Suitable Model
• A model is a mathematical representation of a problem situation. The mathematical model is in the form of expressions and
equations that replicate the problem. For example, the total profit from a given number of products sold can be determined by
subtracting selling price and cost price and multiplying the number of units sold. Assuming selling price, sp as Rs. 40 and cost price,
cp as Rs. 20, the following mathematical model expresses the total profit, tp earned by selling number of unit x.
• TP = (SP – CP) x
• = (40 – 20) x
• TP = 20 x
Now, this mathematical model enables us to identify the real situation by understanding the model. The models can be used to
maximize the profits or to minimize the costs. The applications of models are wide, such as:
• l Linear Programming Model
• l Integer Programming
• l Sensitivity Analysis
• l Goal Programming
• l Dynamic Programming
• l Non Linear Programming
• l Queuing Theory
• l Inventory Management Techniques
• PERT/CPM (Network Analysis)
• l Decision Theory
• l Games Theory
• l Transportation and Assignment Models.
• Acquiring the Input Data
• Accurate data for input values are essential. Even though the model is well constructed, it is important that the
input data is correct to get accurate results. Inaccurate data will lead to wrong decisions.
• Solving the Model
• Solving is trying for the best result by manipulating the model to the problem. This is done by checking every
equation and its diverse courses of action. A trial and error method can be used to solve the model that enables
us to find good solutions to the problem.
• Validating the Model
• A validation is a complete test of the model to confirm that it provides an accurate representation of the real
problem. This helps us in determining how good and realistic the solution is. During the model validation
process, inaccuracies can be rectified by taking corrective actions, until the model is found to be fit.
• Implementing the Results
• Once the model is tested and validated, it is ready for implementation. Implementation involves
translation/application of solution in the company. Close administration and monitoring is required after the
solution is implemented, in order to address any proposed changes that call for modification, under actual
working conditions.
Definitions of Operations Research

• 1. O.R. is a scientific method of providing executive departments with a quantitative basis for decisions
regarding the operations under their control.
• -Morse and Kimbal (1946)
• 2. O.R. is a scientific method of providing executive with an analytical and objective basis for decisions.
• -P.M.S. Blackett (1948)
• 3. O.R. is the application of scientific methods, techniques and tools to problems involving the operations of
system so as to provide these in control of the operations with optimum solutions to the problem.
• -Churchman, Acoff, Arnoff (1957)

On the basis of all above opinions, Operations Research can be defined in more general and comprehensive way
as:
• “Operation research is a branch of science which is concerned with the application of scientific methods and
techniques to decision making problems and with establishing the optimal solutions".
• As the discipline of operations research grew, numerous names such as Operations Analysis,
Systems Analysis, Decision Analysis, Management Science, Quantitative Analysis, Decision
Science were given to it. This is because of the fact that the types of problems encountered are
always concerned with ‘effective decision’.
SCOPE OF OPERATIONS RESEARCH
ADVANTAGES OF OPERATIONS RESEARCH
• Following are certain advantages of Operations Research (OR):
• Operations Research helps decision –maker to take better and quicker decisions. It helps decision –maker to
evaluate the risk and results of all the alternative decisions. So, it improves the quality of decisions and makes the
decisions more effective.
• Operation Research helps, in preparing future managers as it provides in-depth knowledge about a particular
action.
• Operations Research develop models, which provides logical and systematic approach for understanding, Solving
and controlling a problem.
• Operations research reduces the chances of failure as it provides many alternatives for one problem, which helps
the management to choose the best decision. Even managers can evaluate the risks associated with each solution and
can decide whether they want to go with the solution or not.
• It helps users in optimum use of resources. For example, linear programming techniques in Operations Research
suggest most effective methods and efficient ways of optimality.
• It helps in finding the limitations and scope of an activity.
• Using this information, he can measure the performance of employees and can compare it with the standard
performance. It modifies mathematical solutions before these are applied. Managers may accept or modify the
mathematical solutions obtained using Operations Research techniques.
• It helps suggest alternative solutions for the same optimum profit if the management wants so.
LIMITATIONS OF OPERATIONS RESEARCH

• Formulation of mathematical models may take into account all possible factors for defining a real-life
problem and hence is difficult. As a result, the help of computers is required for the large number of
cumbersome computations for such problems. This discourages small companies and other organisations from
using O.R. techniques.
• Unquantifiable factors: Some problems may involve a large number of intangible factors such as human
emotions, human relationship, etc. which cannot be quantified. Hence, the best solution cannot be determined
for such problems because such factors have to be excluded.
• Dependence on experts: A specialist, who may be a mathematician or a statistician, is needed to understand
the formulation of models, find solutions and recommend their implementation. Managers, who deal with
such problems, may not have such specialization. Managers, who deal with such problems, may not have such
specialization and hence the results may not be optimal.
• Model is abstraction of real-life situations and not the reality.
• Assumptions need to be made about the nature and importance of some factors in order to construct an
Operation Research model.
• A reasonably good solution without the use of Operation Research may be preferred by the management as
compared to a slightly better solution provided by using Operation Research since it is very expensive in
terms of time and money.
FEATURES OF OPERATIONS RESEARCH APPROACH
• OR utilizes a planned approach following a scientific method and an interdisciplinary team, in order to represent
complex functional relationship as mathematical models, for the purpose of providing a quantitative basis for
decision-making and uncovering new problems for quantitative analysis. This definition implies additional
features of OR approach. The broad features of OR approach in solving any decision problem are summarized
as follows:
• Interdisciplinary approach For solving any managerial decision problem often an interdisciplinary teamwork is
essential. This is because while attempting to solve a complex management problem, one person may not have
the complete knowledge of all its aspects such as economic, social, political, psychological, engineering, etc.
Hence, a team of individuals specializing in various functional areas of management should be organized so that
each aspect of the problem can be analysed to arrive at a solution acceptable to all sections of the organization.
• Scientific approach Operations research is the application of scientific methods, techniques and tools to
problems involving the operations of systems so as to provide those in control of operations with optimum
solutions to the problems (Churchman et al.). The scientific method consists of observing and defining the
problem; formulating and testing the hypothesis; and analysing the results of the test. The data so obtained is
then used to decide whether the hypothesis should be accepted or not. If the hypothesis is accepted, the results
should be implemented, otherwise not.
• Holistic approach While arriving at a decision, an operations research team examines the relative importance
of all conflicting and multiple objectives. It also examines the validity of claims of various departments of the
organization from the perspective of its implications to the whole organization.
• Objective-oriented approach An operations research approach seeks to obtain an optimal solution to the
problem under analysis. For this, a measure of desirability (or effectiveness) is defined, based on the
objective(s) of the organization. A measure of desirability so defined is then used to compare alternative courses
of action with respect to their possible outcomes.
A note on systems
• A system is defined as an arrangement of components designed to achieve a particular objective or
objectives according to plan. The components may either be physical or conceptual or both, but they all
share a unique relationship with each other and with the overall objective of the system.
• A solution is feasible if it satisfies all the constraints. It is optimal if, in addition to being feasible, it yields
the best (maximum or minimum) value of the objective function. Though OR models are designed to
optimize a specific objective criterion subject to a set of constraints, the quality of the resulting solution
depends on the degree of completeness of the model in representing the real system.
• Solving the OR Model
• In practice, OR does not offer a single general technique for solving all mathematical models. Instead,
the type and complexity of the mathematical model dictate the nature of the solution method.
• The most prominent OR technique is linear programming. It is designed for models with linear
objective and constraint functions. Other techniques include integer programming (in which the
variables assume integer values), dynamic programming (in which the original model can be
decomposed into smaller more manageable subproblems), network programming (in which the
problem can be modeled as a network), and nonlinear programming (in which functions of the model
are nonlinear). These are only a few among many available OR tools. A peculiarity of most OR
techniques is that solutions are not generally obtained in (formula-like) closed forms. Instead, they are
determined by algorithms.
• An algorithm provides fixed computational rules that are applied repetitively to the problem, with each
repetition (called iteration) attempting to move the solution closer to the optimum. Because the
computations in each iteration are typically tedious and voluminous, it is imperative in practice to use
the computer to carry out these algorithms. Some mathematical models may be so complex that it
becomes impossible to solve them by any of the available optimization algorithms. In such cases, it may
be necessary to abandon the search for the optimal solution and simply seek a good solution using
heuristics or metaheuristics, a collection of intelligent search rules of thumb that move the solution
point advantageously toward the optimum.
More than Just Mathematics

• Because of the mathematical nature of OR models, one tends to think that an OR study is always
rooted in mathematical analysis. Though mathematical modeling is a cornerstone of OR, simpler
approaches should be explored first. In some cases, a “commonsense” solution may be reached
through simple observations. Indeed, since the human element invariably affects most decision
problems, a study of the psychology of people may be key to solving the problem.
Some instances of common sense over
modelling:
• The stakes were high in 2004 when United Parcel Service (UPS) unrolled its ORION software (based on the
sophisticated Traveling Salesman Algorithm) to provide its drivers with tailored daily delivery itineraries. The software
generally proposed shorter routes than those presently taken by the drivers, with potential savings of millions of dollars
a year. For their part, the drivers resented the notion that a machine could “best” them, given their long years of
experience on the job. Faced with this human dilemma, ORION developers resolved the issue simply placing a visible
banner on the itinerary sheets that read “Beat the Computer.” At the same time, they kept ORION-generated routes
intact. The drivers took the challenge to heart, with some actually beating the computer suggested route. ORION was
no longer putting them down. Instead, they regarded the software as complementing their intuition and experience
• Travelers arriving at the Intercontinental Airport in Houston, Texas, complained about the long wait for their baggage.
Authorities increased the number of baggage handlers in hope of alleviating the problem, but the complaints persisted.
In the end, the decision was made to simply move arrival gates farther away from baggage claim, forcing the
passengers to walk longer before reaching the baggage area. The complaints disappeared because the extra walking
allowed ample time for the luggage to be delivered to the carousel.
• In a steel mill in India, ingots were first produced from iron ore and then used in the manufacture of steel bars and
beams. The manager noticed a long delay between the ingots production and their transfer to the next manufacturing
phase (where end products were produced). Ideally, to reduce reheating cost, manufacturing should start soon after the
ingots leave the furnaces. Initially, the problem could be perceived as a line-balancing situation, which could be
resolved either by reducing the output of ingots or by increasing the capacity of manufacturing. Instead, the OR team
used simple charts to summarize the output of the furnaces during the three shifts of the day. They discovered that
during the third shift starting at 11:00 P.M., most of the ingots were produced between 2:00 and 7:00 A.M. Investigation
revealed that third-shift operators preferred to get long periods of rest at the start of the shift and then make up for lost
production during morning hours. Clearly, the third-shift operators have hours to spare to meet their quota. The problem
was solved by “leveling out” both the number of operators and the production schedule of ingots throughout the shift.
• 5. In response to complaints of slow elevator service in a large office building, the OR team initially perceived the
situation as a waiting-line problem that might require the use of mathematical queuing analysis or simulation. After
studying the behavior of the people voicing the complaint, the psychologist on the team suggested installing full-length
mirrors at the entrance to the elevators. The complaints disappeared, as people were kept occupied watching
themselves and others while waiting for the elevator.
• In a study of the check-in counters at a large British airport, a U.S.– Canadian consulting team used
queuing theory to investigate and analyze the situation. Part of the solution recommended the use of
well-placed signs urging passengers within 20 mins of departure time to advance to the head of the
queue and request priority service. The solution was not successful because the passengers, being
mostly British, were “conditioned to very strict queuing behavior.” Hence they were reluctant to move
ahead of others waiting in the queue.
• A number of departments in a production facility share the use of three trucks to transport material.
Requests initiated by a department are filled on a first-comefirst- serve basis. Nevertheless, the
departments complained of long wait for service, and demanded adding a fourth truck to the pool.
Ensuing simple tallying of the usage of the trucks showed modest daily utilization, obviating a fourth
truck. Further investigations revealed that the trucks were parked in an obscure parking lot out of the
line of vision for the departments. A requesting supervisor, lacking visual sighting of the trucks, assumed
that no trucks were available and hence did not initiate a request. The problem was solved simply by
installing two-way radio communication between the truck lot and each department.
• Four conclusions can be drawn from these illustrations:
• 1. The OR team should explore the possibility of using “different” ideas to resolve the situation. The
(common-sense) solutions are rooted in human psychology rather than in mathematical modeling. This
is the reason OR teams may generally seek the expertise of individuals trained in social science and
psychology, a point that was recognized and implemented by the first OR team in Britain during World
War II.
• 2. Before jumping to the use of sophisticated mathematical modeling, a bird’s eye view of the situation
should be adopted to uncover possible nontechnical reasons that led to the problem in the first place. In
the steel mill situation, this was achieved by using only simple charting of the ingots production to
discover the imbalance in the third-shift operation. A similar simple observation in the case with the
transport trucks situation also led to a simple solution of the problem.
• 3. An OR study should not start with a bias toward using a specific mathematical tool before the use of
the tool is justified. For example, because linear programming is a successful technique, there is a
tendency to use it as the modeling tool of choice. Such an approach may lead to a mathematical model
that is far removed from the real situation. It is thus imperative to analyze available data, using the
simplest possible technique, to understand the essence of the problem. Once the problem is defined, a
decision can be made regarding the most appropriate tool for the solution. In the steel mill problem,
simple charting of the ingots production was all that was needed to clarify the situation.
• 4. Solutions are rooted in people and not in technology. Any solution that does not take human behavior
into consideration is apt to fail. Even though the solution of the British airport problem may have been
mathematically sound, the fact that the consulting team was unaware of the cultural differences
between the United States and Britain resulted in an unimplementable recommendation (Americans
and Canadians tend to be less formal). The same viewpoint can, in a way, be expressed in thUPS case.
ADVANTAGES OF MATHEMATICAL MODELLING
• The advantages of mathematical modelling are many:
• (a) Models exactly represent the real problem situations.
• (b) Models help managers to take decisions faster and more accurately.
• (c) Models save valuable resources like money and time.
• (d) Large and complex problems can be solved with ease.
• (e) Models act as communicators to others by providing information and impact in changing conditions.
• The key to model-building lies in abstracting only the relevant variables that affect the criteria of the
measures-of-performance of the given system and in expressing the relationship in a suitable form.
However, a model should be as simple as possible so as to give the desired result. On the other
hand, oversimplifying the problem can also lead to a poor decision. Model enrichment is done by
changing value of variables, and relaxing assumptions. The essential three qualities of any model
are:
• Validity of the model – model should represent the critical aspects of the system/problem under
study,
• Usability of the model – a model can be used for the specific purposes, and
• Value of the model to the user.
• Besides these three qualities, other consideration of interest are, (i) cost of the model and its
sophistication,
• (ii) time involved in formulating the model, etc.
OPERATIONS RESEARCH MODELS IN PRACTICE
• Allocation models Allocation models are used to allocate resources to activities so as to optimize
measure of effectiveness (objective function). Mathematical programming is the broad term for the OR
techniques used to solve allocation problems. If the measure of effectiveness such as profit, cost, etc., is
represented as a linear function of several variables and limitations on resources (constraints) are
expressed as a system of linear equalities or inequalities, the allocation problem is classified as a linear
programming problem. But, if the objective function or all constraints cannot be expressed as a system of
linear equalities or inequalities, the allocation problem is classified as a non-linear programming
problem. When the value of decision variables in a problem is restricted to integer values or just zero-
one values, the problem is classified as an integer programming problem or a zero-one programming
problem, respectively. A problem having multiple, conflicting and incommensurable objective functions
(goals) subject to linear constraints is called a goal programming problem. If values of decision
variables in the linear programming problem are not deterministic, then such a problem is called a
stochastic programming problem. If resources (such as workers, machines or salesmen) have to be
assigned to perform a certain number of activities (such as jobs or territories) on a one-to-one basis so as
to minimize total time, cost or distance involved in performing a given activity, such problems are
classified as assignment problems. But if the activities require more than one resource and conversely
if the resources can be used for more than one activity, the allocation problem is classified as a
transportation problem.
• Inventory models These models deal with the problem of determination of how much to order at a
point in time and when to place an order. The main objective is to minimize the sum of three conflicting
inventory costs: the cost of holding or carrying extra inventory, the cost of shortage or delay in the
delivery of items when it is needed and the cost of ordering or set-up. These are also useful in dealing
with quantity discounts and selective inventory control.
• Waiting line (or Queuing) models These models establish a trade-off between costs of providing
service and the waiting time of a customer in the queuing system. A queuing model describes: Arrival
process, queue structure and service process and solution for the measure of performance – average
length of waiting time, average time spent by the customer in the line, traffic intensity, etc., of the waiting
system.
• Competitive (Game Theory) models These models are used to characterize the behaviour of two or
more competitors (called players) competing to achieve their conflicting goals. These models are
classified according to several factors such as number of competitors, sum of loss and gain, and the type
• Network models These models are applied to the management (planning, controlling and scheduling)
of large-scale projects. PERT/CPM techniques help in identifying delay and project critical path. These
techniques improve project coordination and enable the efficient use of resources. Network methods
are also used to determine time-cost trade-off, resource allocation and help in updating activity time.
• Sequencing models These models are used to determine the sequence (order) in which a number
of tasks can be performed by a number of service facilities such as hospital, plant, etc., in such a way
that some measure of performance (such as total time to process all the jobs on all the machines) is
optimized.
• Replacement models These models are used to calculate optimal time to replace an equipment
when either its efficiency deteriorates with time or fails immediately and completely.
• Dynamic programming models These models are used where a problem requires optimization of
multistage (sequence of interrelated decisions) decision processes. The method starts by dividing a
given problem into stages or sub-problems and then solves those sub-problems sequentially until the
solution to the original problem is obtained.
• Markov-chain models These models are used for analyzing a system which changes over a period
of time among various possible outcomes or states. That is, these models describe transitions in terms
of transition probabilities of various states.
• Simulation models These models are used to evaluate alternative courses of action by
experimenting with a mathematical model of the problems with random variables. Thus, repetition of
the process by using the simulation model provides an indication of the merit of alternative course of
action with respect to the decision variables.
• Decision analysis models These models deal with the selection of an optimal course of action
given the possible payoffs and their associated probabilities of occurrence. These models are broadly
applied to problems involving decision-making under risk and uncertainty
• COMPUTER SOFTWARE FOR OPERATIONS RESEARCH
• Many real-life OR models require long and complex mathematical calculations. Thus, computer
software packages that are used to do these calculations rapidly and effectively have become a part
of OR approach to problem solving. Computer facilities such as spread sheets or statistical and
mathematical software packages that make such analysis readily available to a decision-maker.