Lecture # 01
OPERATIONS RESEARCH (OR)
Recommended Books and resources
1. Introductory Management Science by F. J. Gould , G.
D. Eppen , C. P. Schmidt
2. Introduction to Management by Bernard. W. Taylor
3. http://www.zeepedia.com
4. Operations Research by A. M. Natarajan, P.
Balasubramanie, A. TAmilarasi
5. Introduction to operations research by Hiller /
Lieberman
Introduction
It is the sub-field of mathematics.
Operations
The activities/processes carried out in an
organization.
Research
The process of observing and investigating for
something new or better.
Introduction
Operations Research : How to better coordinate and
conduct the activities within an organization.
OR is the representation of real world systems by
mathematical models together with the use of
quantitative methods for solving such models with a
view to optimizing
Introduction
Management science is the application of a scientific
approach to solving management problems in order to
help managers make better decisions.
As implied by this definition, management science
encompasses a number of mathematically oriented
techniques.
Management science can be used in a variety
of organizations to solve many different types of
problems
INTRODUCTION
It is a specialized discipline for business decision-making.
OR represents the study of optimal resource allocation.
The goal of OR is to provide rational bases for decision making by
seeking to understand and structure complex situations, and to utilize
this understanding to predict system behavior and improve system
performance.
Using analytical and numerical techniques to develop and manipulate
mathematical models of organizational systems that are composed of
people, machines, and procedures.
OR involves solving problems that have complex structural,
operational and investment dimensions, involving the allocation and
scheduling of resources.
Terminology
• The British/Europeans refer to “Operational Research",
the Americans to “Operations Research" - but both are
often shortened to just "OR".
• Another term used for this field is “Management
Science" ("MS"). In U.S. OR and MS are combined
together to form "OR/MS" or "ORMS".
• Yet other terms sometimes used are “Industrial
Engineering" ("IE") and “Decision Science" ("DS").
Terminology
In simple terms, it is described as
“the science of better”.
Management Science/OR
Approach to Problem Solving
Observation/Identification Of The
Problem
 “Right solution” can not be
obtained from the “wrong
problem.
 Management scientist/OR
specialist is trained to
identify the problems.
 This phase helps examine
the problem that exist in an
organization.
Problem Definition
Problem must be clearly and concisely defined
Improperly defining a problem can easily result in
◦ no solution or
◦ an inappropriate solution
Model Construction
A management science Model is an abstract
representation of an existing problem situation.
Most frequently a management science model
consists of a set of mathematical relationships.
Model Construction
Consider a business firm that sells a product. The product costs
$5 to produce and sells for $20.
A model that computes the total profit that will accrue from the
items sold is
Independent
variable
Z = $20x - $5x
Parameters
Dependent
variable
In this equation, x represents the number of units of the product
that are sold
Z Represents the total profit that results from the sale of the
product
Model Construction
The equation as a whole is known as a functional
relationship.
A model is a functional relationship that includes
variables, parameters, and equations
Model Solution
Once models have been constructed in
management science, they are solved using the
management science techniques.
When we refer to model solution, we also mean
problem solution.
Implementation
Implementation is the actual use of a model once it has
been developed.
This is a critical but often over-looked step in the process.
If the management science model and solution are not
implemented, then the efforts and resources used in their
development will have been wasted.
OR is an On-going Process
Once the steps described above are complete,
it
does
not
necessarily
mean
that
OR
process is completed.
The model results and the decisions based on the results
provide feedback to the original model.
The original OR model can be modified to
test
different
conditions
and
decisions that might occur in the future.
SCOPE
Finance
Budgeting and
investments
Purchasing
Procurement
and Exploration
Marketing
Management
Production
Management
Personal
Management
SCOPE
Better
Control
Better Coordination
Better
Decisions
Better
Systems
Role in
Managerial
Decision
Making
Classification of Management Science Techniques
21
Chapter 1Management
Science
Characteristics of Modeling Techniques





Linear Mathematical Programming - clear objective;
restrictions on resources and requirements; parameters
known with certainty.
Probabilistic Techniques - results contain uncertainty.
Network Techniques - model often formulated as diagram;
deterministic or probabilistic.
Forecasting and Inventory Analysis Techniques probabilistic and deterministic methods in demand
forecasting and inventory control.
Other Techniques - variety of deterministic and
probabilistic methods for specific types of problems.
22
Chapter 1Management
Science
Linear Programming Model
(LP Model)
Linear programming is a model that consists of linear
relationships representing a firm’s decision(s),given an
objective and resource constraints.
The linear programming technique derives its name from
the fact that the functional relationships in the
mathematical model are linear and the solution
technique consists of predetermined mathematical
steps—that is, a program
Linear Programming Model
(LP Model)
A relationship of direct proportionality that, when
plotted on a graph, traces a straight line.
In linear relationships, any given change in
an independent variable will always produce a
corresponding change in the dependent variable
Model Formulation
A linear programming model consists of certain
common components and characteristics.
The model components include decision variables, an
objective function, and model constraints.
Decision variables are mathematical symbols that
represent levels of activity by the firm. For example, an
electrical manufacturing firm desires to produce radios,
toasters, and clocks, where x, y and z are symbols
representing unknown variable quantities of each item
Model Formulation
The objective function is a linear mathematical
relationship that describes the objective of the firm in
terms of the decision variables
The model constraints are also linear relationships of
the decision variables; they represent the restrictions
placed on the firm by the operating environment. The
restrictions can be in the form of limited resources or
restrictive guidelines.
Model Formulation: Example Case Study
Beaver Creek Pottery Company is a small crafts operation run by a
Native American tribal council. The company employs skilled
artisans to produce clay bowls and mugs with authentic Native
American designs and colors.
The two primary resources used by the company are special
pottery clay and skilled labor. Given these limited resources, the
company desires to know how many bowls and mugs to produce
each day in order to maximize profit. This is generally referred to
as a product mix problem type. This scenario is illustrated in
Figure 2.1.
Example Case Study
The two products have the following resource
requirements for production and profit per item
produced (i.e., the model parameters):
Example Case Study
There are 40 hours of labor and 120 pounds of clay
available each day for production. Formulate LP
model for this problem.
STEP 1: Define the decision
variables
The two decision variables represent the number
of bowls and mugs to be produced on a daily basis.
The quantities to be produced can be represented
symbolically as
let x1 be the number of bowls to produce
let x2 be the number of mugs to produce
STEP 2: Formulating the Objective
Function
The objective of the company is to maximize total profit.
The company’s profit is the sum of the individual profits
gained from each bowl and mug.
Maximize Z = 40x1 + 50x2
STEP 3: Formulating the Constraints
The total labor used by the company is the sum of the
individual amounts of labor used for each product.
However, the amount of labor represented by is limited
to 40 hours per day;
thus, the complete labor constraint is
1x1 + 2x2 <= 40 ……….. (1)
STEP 3: Formulating the Constraints
The amount of clay used daily for the production of bowls is 4x1
pounds;
The amount of clay used daily for mugs is 3x2
Given that the amount of clay available for production each day
is 120 pounds, the clay/material constraint can be formulated as:
4x1 + 3x2 <= 120
LP Model
The complete linear programming model for this problem
can now be summarized as follows:
Class Exercise :Case Study of Astro
and Cosmo-a product-Mix Problem
A TV company produces two types of TV sets, the Astro and the
Cosmo.
There are two production lines, one for each set. The capacity of
the Astro production line is 70 sets per day. The capacity of the
Cosmo production line is 50 sets per day.
Apart from this, there are two departments A and B, both of which
are used in the production of each set.
In department A picture tubes are produced. In this department
the Astro set requires 1 labor hour and the Cosmo set requires 2
labor hours. Presently in department A maximum of 120 labor
hours per day are available for production of these two types of
sets.
Case Study of Astro and Cosmo-a
product-Mix Problem
In department B the chassis is constructed. In this
department the Astro set requires 1 labor hour and the
Cosmo set requires 1 labor hour as well. Presently in
department B a maximum of 90 labor hours per day are
available for production of these two types of sets.
Case Study of Astro and
Cosmo-a product-Mix Problem
If the profit contributions are $20 and $30 for each
Astro and Cosmo set, respectively, what should be
the daily production? Construct LP Model to
represent this problem.
LP Model Construction
DEPARTMENT A
( Labor Hours)
Astro
Cosmo
Total
Availability
DEPARTMENT B
( Labor Hours)
Capacity of
Production Lines
(Sets/Day)
PROFIT ($)
1
1
70
20
2
1
50
30
120
90
STEP 1: Define the decision variables
The two decision variables represent the number of Astro
and Cosmo TV sets to be produced on a daily basis. The
quantities to be produced can be represented symbolically
as
Let A be the units of Astros TV sets to be produced per
day and
Let C be the units of Cosmos TV sets to be produced per
day.
STEP 2: Formulating the Objective
Function
The objective of the company is to maximize total profit.
The company’s profit is the sum of the individual profits gained
from Astro and Cosmo TV sets.
Maximize Z = 20A + 30C
STEP 3: Formulating the
Constraints
1. Labor Constraint for Department A
1.A +2.C <= 120
2. Labor Constraint for Department B
1.A +1.C <= 90
3. Astro Production Line Constraint
A <=70
4. Cosmo Production Line Constraint
C<=50
LP Model
Maximize Z = 20A + 30C
Subject to:
A +2C <= 120 ….…… (1)
A + C <= 90 …………..(2)
A <= 50………………… (3)
C <= 70…………….…….(4)
A, C >=0 …………………(5)
Case Study of PROTRAC
Incorporation
Case Study of PROTRAC
Incorporation
Company’s Data
1. Company will make a profit of $5000 on each E-9 and
$4000 on each F-9.
2. Each product is put through the machining operations
in both the departments A and B.
3. For next month’s production, these two departments
have 150 and 160 hours of available time
respectively. Each E-9 uses 10 hours in Department A
and 20 hours in department B whereas each F-9 uses
15 hours in department A and 10 hours in
department B.
Table # 1 Machining Data
Company’s Data
4. In order for management to honor an
agreement with the union, the total labor
hours used in the testing of finished
products can not fall more than 10%
below an arbitrary goal of 150 hours. This
testing is performed in third department.
Each E-9 is given 30 hours of testing and
each F-9 is given 10. The data is
summarized in table#2
Table # 2 Testing Data
Company’s Data
5. In order to maintain the current marketing
position , it is necessary to build at least one F-9
for every three E-9s.
6. To fulfill a customer order during the next
month a total of at least 5 E-9s and F-9s must be
produced in any combination.
Construct LP model
production plan.
to
identify
optimal
STEP 1: Define the decision variables
The two decision variables represent the number
of E-9 and F-9 equipments to be produced during
the next month. The quantities to be produced
can be represented symbolically as
Let E be the number of E-9 equipments to be
produced during the next month
Let F be the number of F-9 equipments to be
produced during the next month
STEP 2: Formulating the Objective
Function
The objective of the company is to maximize total
profit.
The company’s profit is the sum of the individual
profits gained from E-9 and F-9 equipments
Maximize Z = 5000E + 4000F
STEP 3: Formulating the
Constraints
1. Constraint for Department A
10E +15F <= 150
2. Constraint for Department B
20E +10F <= 160
3. Testing Constraint
30E +10F >= 135
STEP 3: Formulating the Constraints
4. Product-mix Constraint
F >= 1/3 E Or
E/3 <= F
E <= 3F
E- 3F <= 0
5. Combination Constraint
E + F >= 5
Final LP Model
Maximize Z = 5000E + 4000F
Subject to:
10E +15F <= 150 ….…… (1)
20E +10F <= 160 ………. (2)
30E +10F >= 135 ……… (3)
E- 3F <= 0….………….…….(4)
E + F >= 5……………………(5)
E, F >=0 ……………………..(6)
Blending Gruel: A Blending Problem
A 16-ounce can of food must contain
proteins, carbohydrates, and fat in at least
the following amounts: protein, 3 ounces;
carbohydrate, 5 ounces; fat, 4 ounces. Four
types of gruel are to be blended together in
various proportions to produce a least-cost
can of dog food satisfying these
requirements. The contents and prices for 16
ounces of the gruel are given below:
Table # 1: Gruel Blending Data
Formulate this gruel Blending Problem as a linear program.
STEP 1: Define the decision variables
Since there are 4 gruels to blended in such a way so as to
fulfill the regular diet requirements while minimizing the
cost, therefore the 4 decision variables represent the
proportion of each gruel in a 16 ounce can of food.
◦ Let x1 denote the proportion of gruel 1 in a 16-ounce can of food.
◦ Let x2 denote the proportion of gruel 2 in a 16-ounce can of food.
◦ Let x3 denote the proportion of gruel 3 in a 16-ounce can of food.
◦ Let x4 denote the proportion of gruel 4 in a 16-ounce can of food.
STEP 2: Formulating the Objective
Function
The objective of the company is to formulate a least
cost can of food satisfying the given requirements.
Minimize Z = 4x1 + 6x2 + 3x3 + 2x4
STEP 3: Formulating the Constraints
1. Constraint for Protein contents
3x1 + 5x2 + 2x3 + 3x4 >= 3
2. Constraint for Carbohydrates
7x1 + 4x2 + 2x3 + 8x4 >= 5
3. Constraint for Fat contents
5x1 + 6x2 + 6x3 + 2x4 >= 4
4. Constraint for ensuring formulation of 1-can of food (16ounces)
x1 + x 2 + x 3 + x 4 = 1
Final LP Model
Minimize Z = 4x1 + 6x2 + 3x3 + 2x4
Subject to:
3x1 + 5x2 + 2x3 + 3x4 >= 3 ….…… (1)
7x1 + 4x2 + 2x3 + 8x4 >= 5…….... (2)
5x1 + 6x2 + 6x3 + 2x4 >= 4 ……… (3)
x1 + x2 + x3 + x4 = 1………….…….(4)
x1 ,x2 , x3 , x4 >=0 ………………….(5)
Security Force Scheduling: A Scheduling
Problem
The personnel manager must schedule a security
force in order to satisfy staffing requirements
shown below.
A Scheduling Problem
Officers work an eight hours shift and there
are six such shifts each day. The starting
and ending time for each of the 6 shifts is
also given below. The personnel manager
wants to determine how many officers need
to work each shift in order to minimize the
total number of officers employed while still
satisfying the staffing requirements.
Table # 2: Shift Schedule
STEP 1: Define the decision variables
Let x1 = number of officers who work in shift 1
Let x2 = number of officers who work in shift 2
Let x3 = number of officers who work in shift 3
Let x4 = number of officers who work in shift 4
Let x5 = number of officers who work in shift 5
Let x6 = number of officers who work in shift 6
Or
Let xi = Number of officers who work in shift I
Where i = 1, ..., 6
STEP 2: Formulating the Objective
Function
The objective of the Personnel manager is to minimize
the total number of officers working in all the shifts
while still satisfying the given requirements.
The total number of officers is the sum of the officers
working in each shift. Therefore:
Min z = x1 + x2 + x3 + x4 + x5 + x6
(Total number of officers employed)
STEP 3: Formulating the Constraints
TIME INTERVAL
SHIFT
12:00A.M
to
4:00 A.M
4:00A.M
to
8:00 A.M
1
x1
x1
2
3
4
x2
8:00A.M
to
NOON
x2
x3
NOON
to
4:00 P.M
4:00 P.M
to
8:00 P.M
x3
x4
x4
5
x5
6
x6
Minimum
Requirement
5
8:00 P.M
to
12.00 A.M
x5
x6
7
15
7
12
9
STEP 3: Formulating the
Constraints
1.
x6 + x 1  5
(12am-4am)
2.
x1 + x 2  7
(4am-8am)
3.
x2 + x3  15
(8am-noon)
4.
x3 + x 4  7
(noon-4pm)
5.
x4 + x5  12
(4pm-8pm)
6.
x5 + x 6  9
(8pm-12am)
7.
xi  0, i = 1, ..., 6
(Non-negativity)
Final LP Model
Min z = x1 + x2 + x3 + x4 + x5 + x6
Subject to:
x6 + x1  5
(12am-4am)
x1 + x 2  7
(4am-8am)
x2 + x3  15
(8am-noon)
x3 + x 4  7
(noon-4pm)
x4 + x5  12
(4pm-8pm)
x5 + x 6  9
(8pm-12am)
xi  0, i = 1, ..., 6
(Non-negativity)
Feasible Solution
A feasible solution to a linear program is a
solution that satisfies all the model
constraints
including
non-negativity
constraint.
Optimal Solution
Definition: An optimal solution to a linear
program is a feasible solution with the
largest objective function value (for a
maximization problem) and minimum value
(for a minimization problem). The value of
the objective function for the optimal
solution is said to be the value of the linear
program.
InFeasible Solution
An Infeasible solution to a linear
program is a one that violates at least
one model constraint including nonnegativity constraint.
Example
Maximize Z = 5000E + 4000F
Subject to:
10E +15F <= 150 ….…… (1)
20E +10F <= 160 ………. (2)
30E +10F >= 135 ……… (3)
E- 3F <= 0….………….…….(4)
E + F >= 5……………………(5)
E, F >=0 ……………………..(6)
Trial points
1.
Let E= 6, F=5
2.
Let E= 5, F=4
3.
Let E= 6, F=4
Model Solution
1.
Infeasible solution
2.
Feasible Solution
3.
Feasible Solution