Operations Research: Making Better Decisions

Questions and Answers

In the given equation (1), what does L.H.S. represent?

A function of arithmetic and geometric means

What is the relation between arithmetic mean and geometric mean?

Arithmetic mean is always greater than the geometric mean

What is the result when taking the geometric mean of two numbers?

The result may be greater, smaller, or equal to either number

What does the inequality 𝑥12 + 𝑦12 ≤ 𝑟 2 represent?

Sum of squares of coordinates is less than or equal to 𝑟 2 Signup and view all the answers

What does 𝜆𝑥1 , 𝑦1 + 1 − λ 𝑥2 , 𝑦2 ∈ 𝐺 represent?

A linear combination of points in set G with coefficients λ and (1-λ) Signup and view all the answers

In the primal simplex method, when is there an unbounded solution when maximizing the objective function $Z$?

When one of the coefficients of the non-basic variable in $Z$-equation is negative, while all the corresponding entries in the constraint matrix are non-positive Signup and view all the answers

In the primal simplex method, when is there an unbounded solution when minimizing the objective function $Z$?

When one of the coefficients of the non-basic variable in $Z$-equation is positive, while all the corresponding entries in the constraint matrix are non-positive Signup and view all the answers

In artificial starting solution for the primal simplex method, what are surplus variables introduced for?

To transform a problem into standard form by converting inequality constraints to equality constraints Signup and view all the answers

In a linear programming problem, what does it mean when constraints have no slack variables?

The problem is already formulated in standard form Signup and view all the answers

In a linear programming problem, what does it mean when constraints have surplus variables?

There are excess resources or capacity available for each constraint Signup and view all the answers

Study Notes

What is Operations Research?

Operations Research is the scientific study of operations to make better decisions, maximizing benefits and minimizing effort and time.
It is used to analyze complex real-life problems to improve or optimize performance.

History of Operations Research

Operations Research originated during World War II, when the UK used it to win the war by effectively using limited military resources.
It was used to study strategic and tactical problems associated with air and land defense of the country.

Convex Sets and Convex Functions

Definition of Line

A line is defined as a set of points 𝐿 = 𝑥 ȁ𝑥 = 𝜆𝑥1 + 1 − 𝜆 𝑥2 , 𝜆𝜖𝑅 that passes through two points 𝑥1 and 𝑥2 in 𝑆 ⊆ 𝑅𝑛.

Definition of Line Segment

A line segment is a set of points 𝐿 = 𝑥 ȁ𝑥 = 𝜆𝑥1 + 1 − 𝜆 𝑥2 , 0 ≤ 𝜆 ≤ 1 that lies on the line between two points 𝑥1 and 𝑥2 in 𝑆 ⊆ 𝑅𝑛.

Definition of Convex Set

A set 𝐾 ⊆ 𝑅𝑛 is convex if for each 𝑥1, 𝑥2 ∈ 𝐾, then 𝑥 ∈ 𝐾, where 𝑥 = 𝜆𝑥1 + 1 − 𝜆 𝑥2, 0 ≤ 𝜆 ≤ 1.

Example of Convex Set

The set 𝐺 = { 𝑥1, 𝑥2 : 𝑥12 + 𝑥22 ≤ 𝑟2 } is convex.

Analytic Solution of Linear Programming Problems

Types of Solutions in Primal Simplex Method

Unique Optimal Solution: obtained when all coefficients of non-basic variables in the 𝑍-equation are positive or negative.
Non-Unique Optimal Solution (infinite number of solutions): obtained when at least one coefficient of non-basic variables in the 𝑍-equation is zero, while all other coefficients are non-negative or non-positive.