1. , Introduction to Operations Research, 7th ed. McGraw Hill,
Frederick S. Hillier and Gerald J. Lieberman

2. History of Operations Research:
Chapter 1 Introduction
History of Operations Research:
Research on Radar: This work involved the closest possible cooperation between the scientists and the officers and men of the (English) Royal Air Force, so that the best tactical operations of both equipment and men, air and ground crews, could be achieved.

3. 1938
A.P. Rowe was in charge of the scientific group at Bawdsey, refereed to it as "Operational Research". (Bawdsey is the birthplace of Operations Research.)
1942
U.S. captain W. D. Baker, an anti-submarine warfare officer with Atlantic Fleet, requested the establishment of an Anti-Submarine Warfare Operations Research Group(ASWORG). To lead ASWORG, later named Operations Research Group and attached to the Headquarters of the Commander-in-Chief, U.S. Navy, Philip, M. Morse, a physicist,

4. was recruited from Massachusetts Institute of Technology to be project supervisor, and William Shockley, later to win a Nobel Prize for his work on the transistor, was brought from he Bell Laboratories to be director of research.
Blackett early 1941 memorandum emphasized that the work of operational research was the "scientific analysis of operations,". Its objective is to assist the finding of means to improve the efficiency of air operations in progress or planned for the future. To do this, past operations are studied to determine the facts; theories are elaborated to explain the facts; and finally the facts and theories are used to make predictions about future operations . . . 

5. What is Operations Research/ Management Sciences?
Research on (military) operations, 1935  
Management Sciences (synonym)  
What is Operations Research/ Management Sciences?
1. The focus of the Operations Research is on applications of mathematics and logic in research and decision-making on operations, as well as problem solving in all types of public and private organization business, industry, government and state enterprises in order to bring about optimal outcomes in organizational operations.

6. 2. “Scientific approach to decision making that involves the operations for organized systems. O.R. is concerned with optimal decision making in and modeling of deterministic and probabilistic systems that originate from real life,” – Hillier & Lieberman, Introduction to Operations Research, 7th Ed., Holden-Day, 1996
3. “Operations Research is the application of scientific methods to decision problems. It has found wide use and acceptance in all areas of business, government and industry.” – Saul L. Gass, College of Business & Management, University of Maryland, 1979.

7. 4. “The use of analytic methods adapted from mathematics for solving operational and business problems” – Computer Dictionary, Charles J. Sippl and Charles P. Sippl, Howard W. Sams & Co., Inc., Indianapolis, 1978.
5. “A scientific method of providing executive department with a quantitative basis for decisions making operations under their control.” – Morse & Kimball, Methods of Operations Research, Columbia University Press for office of Naval Research, 1943 (9th printing, 1963).

8. 6. “A branch of applied mathematics wherein the application is to the decision making process,” – Donald Gross, Department of Operations Research, The George Washington University, 1979.
7. Operations research employs mathematical models to suggest how best to operate and coordinate the activities within an organization. The tools of operations research are applied in industry, commerce, government, the military, education, health - in fact in virtually any area of human activity.

9. The Origins of Operations Research:
8. Operations Research (OR) is the study of how to form mathematical models of complex engineering and management problems and how to analyze them to gain insight about possible solutions. - Ronald L. Rardin, Optimization in Operations Research, prentice-Hall, Inc., 1998
The Origins of Operations Research:
1. The World War II; Its object is to assist the finding of means to improve the war operations in progress or planned for the future. To do this past studied to determine the facts; theories are elaborated to explain finally the facts and theories are used to make predictions about future operations.

10. 2. Industrial revolution - increasing complexity (size) and specification in (creating new problems)
3. Computer Revolution
4. OR is applied to problems that concern how to conduct and coordinate the activities within an organization.
5. Scientific method; scientific research into the fundamental properties of practical management of the organization; OR uses the method of understand and explain phenomena of operating system.

11. The Impact of Operations Research:
1. International Federation of Operations Research Societies (IFORS) -more than 30 countries, January,1, 1959.
2. Operations Research Society of America ORSA, founded May 26-27, 1952.
3. The Institute of Management Sciences TIMS, founded 1953.
4. The Institute of Operations Research and Management Sciences INFORMS: ORSA and TIMS are merged in 1995.
5. Operational Research Quarterly (Journal of Operational Research Society), the appearing in March  

12. Applications of Operations Research:
Team approach

14. THE METHODS AND THEORIES
THE DISCIPLINE
THE METHODS AND THEORIES
Physical Sciences
Mathematics
Political Sciences
Social Sciences
Business Administration
Industrial Engineering
Behavior Science
Economics
Computer Science
. . .
Decision Theory
Mathematical Programming Queuing Theories
Scheduling Theory
Reliability Theory
Probability& Statistics Stochastic Process
Simulation
Inventory Theory
Network Theory
. . . ↘ ↙
Operations Research ↓
The Applications ↓
Education, Manufacturing, Heath, Finance, Energy and Utilities, Transportation, Environmental, Military, Forest Management

15. Chapter 2 Overview of The
Operations Research Modeling Approach
Define the Problem
of Interest
Gathering
Relevant Data
revise
Formulating a
Mathematical
Model
Developing
Computer
Algorithm
Test the
Model
Applications
of the Model
Implement

16. Define the problem of interest: Determine the appropriate objectives, constraints, interrelationships between the area to be studied and other areas of the organization, possible alternative courses of action, time limits for making a decision,
Gathering relevant data: e.g. install a new computer-based management information system to collect the necessary data.
Formulate a mathematical model: Build a mathematical model for abstracting the essence of the subject of inquiry, showing interrelationships, and facilitating analysis.
Developing computer algorithm: Develop a (computer-based) procedure for deriving solutions to the problem from this model or using one of a number of readily available software packages. Samuel Eilon: "optimizing is the science of the ultimate; satisfying is the art of the feasible."

18. Test the model: The early version of a large mathematical model inevitably contains many flaws. e.g. some interrelationships have not been incorporated into the model, some parameters have not been estimated correctly.
Application of the model: If the model is to be used repeatedly, then it is necessary to install a well-documented system for applying the model as prescribed by management.
Implementation: The success of the implementation phase depends a great deal upon the support of both top management and operating management.

19. Creative Thinking and Problem Solving
Imagination is more important than knowledge, for knowledge is limited Whereas imagination embrace the entire world, stimulating progress giving birth to evolution.
Albert Einstein

21. Chapter 3 Introduction to Linear Programming
Linear Programming: linear (linear functions) programming (planning of activities), Programming problems are concerned with efficient use of limited resources to meet desired objectives.
Solution Methods: Graphical Method, Simplex Method, Interior-Point Method
Historical Review:
1938 – Soviet mathematician and economist L. V. Kantorovich formulated a
linear programming problem dealing with the organization and
planning of production.
1941 – The transportation problem are posed by Hitchcock.
1945 – The diet problem (dealt with the optimization of linear function subject
to linear constraint) is posed by Stigler.

22. Dantzig, Kantorovich, Beale (Laxenburg, 1976)

23. Koopmans, Dantzig, Kantorovich (Laxenburg, 1976)

24. 1947 – The transportation problem is posed independently by Koopmans.
The initial mathematical statement of the general problem of linear programming was made by Dantzig in 1947 along with the Simplex Method.
The general problem of linear programming was first developed and applied in 1947 by George B. Dantzig, Marshall Wood, and their associates of the U.S. Department of Air Force. At that time this group was called on to investigate the feasibility of applying mathematical and related techniques to military programming and planning problems. This inquiry led Dantzig to propose “that interrelationship between activities of a large organization be viewed as a linear programming type model and the optimizing program determined by minimizing a linear objective function.”
In order to develop and extend these ideas further, the Air Force organized a research group under the title SCOOP (Scientific Computation of Optimum Program). Besides putting the Air Force programming and budgeting problems on a more scientific basis, Project SCOOP's major contribution was the formal development and applications of the linear-programming model.

25. Dantzig, Khachian (Asilomar, 1990)

26. Davidon, Fletcher, Powell

27. These early applications of linear programming methods fell into three major categories: military application generated by Project SCOOP, interindustry economics based on the Leontief input-output model, and problems involving the relationship between zero-sum two-person games and linear programming.
In the past years, the applications of linear programming have been extended and developed, but the main emphasis in linear programming applications has shifted to the general industry area, with many applications found in the social and urban fields.
(Refer “Saul I. Gass, Linear Programming: Methods and Applications, boyd & fraser publishing company, New York,1985, 5th edition.”)
1952 – The first successful solution of a linear programming problem on a high –
speed electronic computer occurred in January 1952, on the National
Bureau of Standards SEAC machine. Since that time, the simplex
algorithm, or variations of this procedure, has been coded for most
intermediate and large general-purpose electronic computers.

28. 1979 – Minty and Klee showed that simplex method’s worst cost is exponential
in the number of variables (but these inputs are rare in practice).
In 1979, the Russian mathematician Leonid Hacijian (“kha-chi-yuan”) building on work by other Russian theorists A. Jurii Levin, N. Z. Shor, D. B. Judin, and A. S. Nemirovskii, proposed and algorithm called the ellipsoid algorithm, which can solve linear program in polynomial time.
1984 – In 1984, Narendra Karmarka, a computer scientist at AT&T Bell Labs
improve the ellipsoid algorithm to run well in practice. Called projective
scaling, Karmarka’s algorithm is polynomial; it outperforms the ellipsoid
algorithm; and it is competitive with simplex algorithm on practical
problems. Since then several new polynomial algorithm for linear
program have been proposed, and new algorithms beat simplex as the
problem size grows.

29. 3.1 Prototype Example:
Example 1: WYNDOR GLASS CO. (product mix type problem)
Plant 1: producing aluminum frames and hardware
Plant 2: producing wood frames
Plant 3: producing glass and assembles the products
Product 1: 8ft glass door with aluminum frame
Product 2: 4ft  6ft double-hung wood-frame window
Plant
Production time(hr.)/batch
Production Time (hr.)
available/week
Product 1
Product 2 1 2 3 4 12 18
Profit/batch
$3,000
$5,000

30. Linear Programming Formulation:
x1 = (# of batches of product 1 produced)/week
x2 = (# of batches of product 2 produced)/week
Z = (total profit in thousands of dollars)/week
Maximize x x2
Subject to x ≦ 4
2x2 ≦ 12
3x x ≦ 18
x1 , x2 ≧ 0.

31. x2 Graphical Method: x1 x1  4 3x1 +2x2  18 2x2  12 ( 2 , 6)
( 4 , 3) x1

32. x2 Graphical Method: x1 Z = 36 = 3x1 +5x2 6 ( 2 , 6) 4 2

33. 3.2 The Linear Programming Model
A Standard Form of the Model:
Maximize Z = c1 x1 + c2 x c n x n
Subject to a11x1 + a12x a1n xn ≦ b1
a21x1 + a22x a2n x n ≦ b2
. . .
. . .
. . .
am1x1 + am2x am n x n ≦ bm
x1, x2 , , x n ≧ 0.
The standard form is useful in exploiting duality relationship.

34. Z = c1 x1 + c2 x2 + . . . + c n x n is the objective function to be maximized
(overall measure of performance)
ai1x1 + ai2x ain xn ≦ bi , i = 1, 2, , m : are functional constraints.
x1, x2 , , x n ≧ 0 : are nonnegativity constraints (or nonnegativity condition).
xj : decision variables (level of activity j, j = 1, , n)
cj : profit coefficients j = 1, , n.
bi : amount of resource available, i = 1, 2, , m.
aij : amount of resource i consumed by each unit of activity, i = 1, 2, , m,
i = 1, 2, , n.
(cj , bi , aij are parameters.)
xj ≧0 : the nonnegative constraints, i = 1, 2, , n. (decision variables)

35. Resource Usage per Unit of Activity
Resource
Resource Usage per Unit of Activity
Amount of Resource Available
Activity n 1 2  m
a a a1n
a a a2n
. . .
am am amn b1 b2 bm

39. Maximize cx subject to Ax ≦ b x ≧ 0.
These technological coefficients form the constraint matrix A given below.
This problem is written in vector form as follows.
Maximize cx
subject to Ax ≦ b
x ≧ 0.

41. Other forms of LP
1. Minimizing the objective function:
Minimize cx
Subject to Ax ≦ b,
x ≧ 0.
2. Inequality constraints:
Maximize cx
Subject to Ax ≧ b,
x ≧ 0.
( or ai1x1 + ai2x ai n x n ≧ bi for some i. )

42. 3. Equality constraints:
ai1x1 + ai2x ai n x n = bi for some i.
4. Unrestricted variables:
-∞ < xi < +∞ for some i.
5. Variables with lower and/or upper bounds:
li ≦ xi ≦ ui for some i.
6. Absolute values:
(a) Maximize ∣x∣+ ∣y∣+ z
Subject to x y ≦1,
2x z ≦ 3,
-∞ < x, y, z < +∞

43. Maximize { minimize { x - y, -2x + 3y }
(b) Maximize x y
Subject to x + 2∣y∣≦ 0
∣3x - y∣≦ 3,
x ≧ 0, -∞ < y < +∞
7. Multiple objectives problem:
Maximize { x - y, -2x +3y }
Subject to x y ≦ 0
3x y ≦ 3,
x, y ≧ 0
8. Minimax (Maximini) problem:
Maximize { minimize { x - y, -2x + 3y }
Subject to -x + 2y ≦ 0
3x - y ≦ 3,
x, y ≧ 0

44. 3.3 Assumptions of linear programming
Terminology for Solutions of the Model: Feasible solution, Feasible region, optimal solution, corner-point feasible (CPF) solution,
The column vector whose ith component is bi, which is referred to as the right-hand-side vector, represents the resources available for allocation. A set of variables x1, x2 , , xn satisfying all the constraints is called a feasible point or a feasible solution. The set of all such points constitues the feasible region (or the feasible space).
3.3 Assumptions of linear programming
1. Proportionality
2. Additivity
3. Divisibility
4. Certainty

45. 3.4 Additional Examples
Example 2. Design of Radiation Therapy
Example 3. Regional Planning
Example 4. Controlling Air Pollution
Example 5. Reclaiming Solid Wastes
Example 6. Personnel Scheduling
Example 7. Distributing Goods through a Distributed Network
. . .

46. Design of Radiation Therapy

49. Regional Planning

50. Formulation as a LP Problem:

54. Example 6. Personnel Scheduling

55. Optimal Solution ( x1, x2, x3, x4, x5 ) = ( 48, 31, 39, 43, 15), Z = 30,610

56. Example 7. Distributing Goods through a Distributed Network

58. The Relationship between Linear Programming and Linear Algebra /Geometric Interpretation of the Linear program
Consider the following standard form of linear program:
Maximize cx
Subject to Ax ≦ b,
x ≧ 0.
Two ways in interpreting a linear program.
(1) One represents a linear program in the space of the decision variables,
called the activity space;
(2) the other represents a linear program in the space of requirements, called
the requirement space.

59. Activity space
The activity space is of dimension n and the feasible region in the activity space is then the intersection of m half-spaces aix ≦ bi and n half-spaces xj ≧ 0, i = 1, ... , m, j = 1, ..., n. Among all such points we wish to find a point with maximal cx value. Note that points with the same objective value z satisfy the equation cx = z. Since z is to be maximized, then the plane cx = z must be moved parallel to itself in the direction that maximizes the objective value most. This direction is c, and hence the plane is moved in the direction c as much as possible.

60. For a maximization problem, all possible cases that may arise are
summarized as follows.
Unique Finite Optimal Solution: If the optimal finite solution is unique, then it occurs at an extreme point. In this case, the feasible region may be bounded or unbounded.
Alternative Finite Optimal Solution: If there are two extreme points that
both are optimal, then any point on the line segment joining them is optimal.
Unbounded Optimal Solution: For a maximization problem the plane cx = z can be moved in the direction c indefinitely which
always intersecting with the feasible region. In this case the optimal
objective is unbounded with value +∞. This can happen only when
the feasible region is unbounded.
Empty Feasible Region. In this case, the system of equations and/or
inequalities defining the feasible region is inconsistent.

61. Summary of Chapter 3
1. Terminology: standard form of linear program, objective function,
constraint, functional constraint, nonnegativity constraint, decision
variable, feasible solution, infeasible solution, feasible region, optimal
solution, corner-point feasible(CPF) solution.
2. What is linear programming?
3. What is the assumptions of linear programming?
4. What is the standard form of linear program? Other forms of linear program?
5. The relationship between linear program and linear algebra.
( What are the corresponding equations of a simplex tableau? )
6. Solution Methods of linear programming: graphical method, simplex
method, Interior-point method.
7. How to formulate a linear program?

62. Duality and Sensitivity analysis
Operations Research
Chapter 4
Linear Programming
Duality and Sensitivity analysis

63. Linear Programming Duality and Sensitivity analysis
Dual Problem of an LPP
Definition of the dual problem
Examples (1,2 and 3)
Sensitivity Analysis
Examples (1 and 2)

64. Dual Problem of an LPP
Given a LPP (called the primal problem), we shall associate another LPP called the dual problem of the original (primal) problem. We shall see that the Optimal values of the primal and dual are the same provided both have finite feasible solutions. This topic is further used to develop another method of solving LPPs and is also used in the sensitivity (or post-optimal) analysis.

65. Definition of the dual problem
Given the primal problem (in standard form)
Maximize
subject to

66. the dual problem is the LPP
Minimize
subject to

67. If the primal problem (in standard form) is
Minimize
subject to

68. Then the dual problem is the LPP
Maximize
subject to

69. We thus note the following:
1. In the dual, there are as many (decision) variables as there are constraints in the primal.
We usually say yi is the dual variable associated with the ith constraint of the primal.
2. There are as many constraints in the dual as there are variables in the primal.

70. 3. If the primal is maximization then the dual is minimization and all constraints are 
If the primal is minimization then the dual is maximization and all constraints are 
In the primal, all variables are  0 while in the dual all the variables are unrestricted in sign.

71. 5. The objective function coefficients cj of the primal are the RHS constants of the dual constraints.
6. The RHS constants bi of the primal constraints are the objective function coefficients of the dual.
7. The coefficient matrix of the constraints of the dual is the transpose of the coefficient matrix of the constraints of the primal.

72. Example1
Write the dual of the LPP
Maximize
subject to

73. Thus the primal in the standard form is:
Maximize
subject to

74. Hence the dual is:
Minimize
subject to

75. Example2
Write the dual of the LPP
Minimize
subject to

76. Thus the primal in the standard form is:
Minimize
subject to

77. Hence the dual is:
Maximize
subject to

78. Example3
Write the dual of the LPP
Maximize
subject to

79. Thus the primal in the standard form is:
Maximize
subject to

80. Hence the dual is:
Minimize
subject to

81. Theorem: The dual of the dual is the primal (original problem).
Proof. Consider the primal problem (in standard form)
Maximize
subject to