peration Research (1).pptx

Operation
Research
An Introduction

Chapter 1
What is
Operations
Research ?

Agenda
 Introduction
 The term of Operations Research (OR)
 Components of OR
 Phases of an OR study
 Operation Research Models
 Queuing and Simulation Models
 Arts of Modeling

 Operations Research is an Art and Science
 It had its early roots in World War II and is flourishing in business and
industry with the aid of computer
 Primary applications areas of Operations Research include forecasting,
production scheduling, inventory control, capital budgeting, and
transportation.
The term of Operations Research (OR)

Operations
• The activities carried out in an organization.
Research
• The process of observation and testing characterized by the
scientific method. Situation, problem statement, model
construction, validation, experimentation, candidate solutions.
Operations Research is a quantitative approach to decision making based on
the scientific method of problem solving.
The term of Operations Research (OR) (Cont.)

Operations Research is the scientific approach to execute decision
making, which consists of:
 The art of mathematical modeling of complex situations
 The science of the development of solution techniques used
to solve these models
 The ability to effectively communicate the results to the
decision maker
What is Operations Research?

What is Operations Research? (Cont.)
1. OR professionals aim to provide rational bases for decision making
by seeking to understand and structure complex situations and to
use this understanding to predict system behavior and improve
system performance.
2. Much of this work is done using analytical and numerical
techniques to develop and manipulate mathematical and
computer models of organizational systems composed of people,
machines, and procedures.

The term of Operations Research (OR)
• It is a scientific approach to determine the optimum (best)
solution to a decision problem under the restriction of
limited resources. Using the mathematical techniques to
model, analyze, and solve the problem.

Basic component of the model
1.Decision Variables
• It is the unknown to be determined from the solution of a model (what does the model seek to
determine). It is one of the specific decisions made by a decision maker (DM).
2.Objective Function
• It is the end result (goal) desired to be achieved by the system. A common object is to maximize
profit or minimize cost. It is expressed as a mathematical function of the system decision
variables.
3.Constraints
• These are the limitation imposed on the variables to satisfy the restriction of the modeled system.
They must be expressed as mathematical functions of the system decision variables (D.V).

Phases Of Operations Research (OR)
1. Definition of the problem includes :
• The description of decision variables (alternatives)
• The determination of the objective of the study
• The specification of the limitations under which the modeled system operates.
2. Model Construction
• Translating the real-world problem into mathematical relationships (the most suitable model to represent
the system, LP, dynamic programming, integer programming,……..)
3.Solution of the model
• Using well-defined optimization techniques.
• An important aspect of model solution is sensitivity analysis.
4. Model validity
• Testing and evaluation of the model. A common method for testing a validity of a model is to compare its
performance with some past data available for the actual system.
5.Implementation of the solution
• Implementation of the solution of validated model involves the translation of the mold's results into
instructions issued in understandable form to the individual

Operation Research Models
• In operations research (OR) a real stories and problems
can be expressed into a mathematical and analytical for
the purpose of selecting the best optimal solution for
problem solving and decision-making that is useful and
needed in the management of organizations.

Operation Research Models Cont.
• Selecting the solutions of Operation Research problems
depends on the following factors :
What is the Objective Function in a certain problem that is
needed to be achieved (Minimize or maximize a specific
objective)?
Defining what are the finite and infinite Solutions or
alternatives that are feasible for achieving the objective
function under a number of constraints ?
Under what Constraints the decision will be taken?
What is an appropriate objective criterion for evaluating the
alternatives?

Applying OR Models In a real-life
Problem 1
Consider the following tickets purchasing problem:
A businessperson has a 5-week commitment traveling between Fayetteville
(FYV) and Denver (DEN). Weekly departure from Fayetteville occurs on
Mondays for return on Wednesdays.
 A regular roundtrip ticket costs $400, but a 20% discount is granted if the
roundtrip dates span a weekend. A one-way ticket in either direction costs 75%
of the regular price. How should the tickets be bought for the 5-week period?
should the tickets be bought for the 5-week period? Consider the following tickets purchasing problem. A
businessperson has a 5-week commitment traveling between Fayetteville (FYV) and Denver (DEN). Weekly
departure from Fayetteville occurs on Mondays for return on Wednesdays. A regular roundtrip ticket costs
$400, but a 20% discount is granted if the roundtrip dates span a weekend. A one-way ticket in either direction
costs 75% of the regular price. How should the tickets be bought for the 5-week period?

Answer of problem 1 :
 First, the objective function will be minimizing the ticket pricing for the whole 5 weeks under a
constraint or restriction that the businessperson should be able to leave from FYD on Monday to
DYN and return to FYD on Wednesday of the same week.
 Second, we have three Feasible Finite solutions that can be applied but the optimal solution will
be evaluated according to minimize the cost of ticket price
 Third the three plausible alternatives will be as follows:
 Alternative 1 :
 Buy five regular round trip FYV-DEN-FYV for departure on Monday and return on Wednesday of
the same week.
 Alternative 1 cost for buying tickets for the 5 weeks for leaving FYD on Monday to DYN and
return to FYD
on Wednesday will be as follow :5 * $400 = $2000 Case 1 has a 5-week commitment traveling
between Fayetteville (FYV) and Denver (DEN). Weekly departure from Fayetteville occurs on Mondays for return on Wednesdays. A
regular roundtrip ticket costs $400, but a 20% discount is granted if the roundtrip dates span a weekend. A one-way ticket in either
direction costs 75% of the regular price. How should the tickets be bought for the 5-week period?

Answer of problem 1 cont :
 Alternative 2 :
 Buy one-way single trip FYV-DEN, four DEN-FYV-DEN that span weekends, and one –
way trip from DEN to FYV
 Alternative 2 cost for buying tickets for the 5 weeks for leaving FYD on Monday to DYN
and return to FYD will be calculated as follows:
 The first single trip is from FYD on Monday to arrive DYN on the same day :
75%* $400 = $300
 The Four round trips that span (include) weekends will be as follows :
1. From DYN on Wednesday to FYD – and from FYD on Monday to DYN
(4*80%* $400) = $1280
So, the total cost for Alternative two will be as ($300 + $1280)= $1880 Case 2

 Alternative 3 :
 Buy one round trip FYV-DEN-FYV to cover Monday of the first week and Wednesday of
the last week and four DEN-FYV-DEN to cover the remaining legs. All tickets in this
alternative span at least one weekend
 Alternative 3 cost for buying tickets for the 5 weeks for leaving FYD on Monday to DYN
and return to FYD will be calculated as follows:
 The first-round trip trip is from FYD on Monday of the first week to arrive DYN on the
same day and return from DYN on Wednesday of the last week ( include weekends):
1*80%* $400 = $320
 The Four round trips that span (include) weekends will be as follows :
(4*80%* $400) = $1280
So, the total cost for Alternative three will be as ($320 + $1280)= $1600 Case 3

 Finally, the selected optimal feasible solution that achieve cost minimization is
alternative 3 with the lowest price is $1600 under a restriction of leaving FYD on
Monday to DYN and return to DYN on Wednesday.
Summary of the problem
Problems1 Finite solutions
5 Regular
Round trips
= $2000
4 round trips
(4 )weekends,
+ 1 single trip
+ 1 single trip=
$1880
4 round trips (4)
weekends,
+ 1 different round
trip with (1 more
weekend) = $1600

Applying OR Models In a real-life
Problem 2
• Consider the following garden problem:
A Homeowner is in the process of starting a backyard vegetable garden. The garden
must take on a rectangular shape to facilitate row irrigation. To keep critters out, the
garden must be fenced. The owner has enough material to build a fence of length L
= 100 ft.
 The goal is to fence the largest possible rectangular area. In contrast with the
tickets example, where the number of alternatives is finite, the number of
alternatives in the present example is infinite; that is, the width and height of the
rectangle can each assume (theoretically) infinity of values between 0 and L. In this
case, the width and the height are continuous variables.
Because the variables of the problem are continuous, it is impossible to find the
solution by exhaustive enumeration. However, we can sense the trend toward the
best value of the garden area by fielding increasing values of width (and hence
decreasing values of height). For example, for L = 100 ft, the combinations (width,
height) = (10, 40), (20, 30), (25, 25), (30, 20), and (40, 10) respectively yield (area)
= (400, 600, 625, 600, and 400), which demonstrates, but not proves, that the
largest area occurs when width = height = L>4 = 25 ft

Problem 2
• First, the objective function will be fencing the largest possible garden with Length (L) =
100 ft under a constraint or restriction that the garden must take on a rectangle shape.
• To demonstrate how the garden problem is expressed mathematically in terms of its two
unknowns, width and height, define
w = width of the rectangle in feet
h = height of the rectangle in feet
• Based on these definitions, the restrictions of the situation can be expressed verbally as
1. Width of rectangle + Height of rectangle = Half the length of the garden fence
2. Width and height cannot be negative.
• These restrictions are translated algebraically as
1. 2 (w + h) = L
2 .w >=0 , h>=0
w >=0 , h>=0

Problem 2
• First, the objective function will be fencing the largest possible garden with Length (L) =
100 ft under a constraint or restriction that the garden must take on a rectangle shape.
• To demonstrate how the garden problem is expressed mathematically in terms of its two
unknowns, width and height, define
w = width of the rectangle in feet
h = height of the rectangle in feet (Length)
• Based on these definitions, the restrictions of the situation can be expressed verbally as
1. Width of rectangle + Height of rectangle = Half the length of the garden fence
2. Width and height cannot be negative.
• These restrictions are translated algebraically as
1. 2 (w + h) = L
2 .w >=0 , h>=0
w >=0 , h>=0

• The only remaining component now is the objective of the problem; namely, maximization of the area of the rectangle.
Let z be the area of the rectangle, then the complete model becomes
Maximize z (area) = w*h
subject to
2(w + h) = L , w, h >= 0
This model can be simplified further by eliminating one of the variables in the objective function using the constraint
equation; that is,
w = L/2 - h
The result is
z = w*h = (L/2 – h)*h = Lh/2 – h*h
The maximization of z is achieved by using differential calculus, which yields the best solution
as h = L 4 = 25 ft. Back substitution in the constraint equation then yields w = L 4 = 25 ft. Thus, the solution calls for
constructing a square-shaped garden.This leads to an infinite number of feasible solutions and, unlike the ticket
purchasing problem, which uses simple price comparisons, the optimum solution is determined using differential
calculus.

Summary of Problem 1&2 :
 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.
 In the ticket purchasing problem, the problem considers three feasible alternatives, with the
third alternative being optimal.
 In the garden problem, a feasible alternative must satisfy the condition w + h = L 2 , with w
and h Ú 0, that is, nonnegative variables.
 This definition leads to an infinite number of feasible solutions and, unlike the ticket
purchasing problem, which uses simple price comparisons, the optimum solution is
determined using differential calculus.

Queuing and Simulation Models
• Deal with the study of waiting lines
• They are not optimization techniques
• They determine measures of performance of waiting lines such as average waiting time in queue, average
waiting time for service.
Example : average waiting time in queue, average waiting time for service,
utilization of service facilities, among others

Continue Queuing and Simulation Models
Queuing Simulation
• Queuing models utilize probability and stochastic
models to analyze waiting line
• Queuing models are purely mathematical
hence are subject to specific assumptions that
limit their scope of application
Queuing
Queuing
• simulation estimates the measures of
performance by “imitating” the behavior of
the real system
• Simulation is flexible and can be used to
analyze practically any queuing situation

The pluses and minuses of simulation
Model
Pluses of Simulation
• flexible and can be used to analyze
practically any queuing situation`
Minuses of Simulation Model
• The process of developing simple
models is more time-consuming
and expensive.
• Furthermore, the execution of the
simulation process, even on faster
computers, is usually slow

Art of Modeling
 The art of modelling involves transforming real-world situations into a mathematical framework
that can be analysed and solved .
 The art of modelling abstract the assumed real-world model from the real situation by
concentrating the problem domain, identifying relevant variables and relationships that control
the behaviour of the real system. The model expresses that in a manner the mathematical
functions represent the assumed real world
 The art of modelling also involves validating and verifying the models by comparing their
outputs to real-world data or performance metrics .This iterative process of model enables
businesses to make informed decisions and continuously improve their operations.

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