mansci

Question 1

definition of linear programming

Answer 1

linear programming is a mathematical method to determining the best possible outcome in a mathematical model with linear relationships. it is often used in optimization problems where the objective is to maximize or minimize a linear function subject to a set of constraints.

Question 2

application of LP

Answer 2

linear programming is widely used in economics, business, engineering, military, transportation, and manufacturing, where optimization of resources is key.

Question 3

components of LP

Answer 3

Decision variables- these are the variables that we need to solve for. they represent the unknowns of the problem.

Objective Function- a linear function that needs to maximize or minimize.

Question 4

Formulating a LP problem

Answer 4

IDENTIFY THE DECISION VARIABLES- determine what quantities need to be solved for.
WRITE THE OBJECT FUNCTION- express the objective (maximization/minimization) as a linear function of the decision variable.
DEFINE THE CONSTRAINTS- identify the limitations or requirements in terms of linear inequalities or equalities
SPECIFY NON-NEGATIVITY RESTRICTION- ensure all decision variables are non-negative

Question 5

GRAPHICAL SOLUTION METHOD (FOR TWO-VARIABLE LP PROBLEM)

Answer 5

STEP 1: plot the constraints as lines on a graph
STEP 2: identify the feasible region (the area where all constraints are satisfied)
STEP 3: plot the objective function and use a method, such as “isoprofit” or “isocost” lines to find the optimal points (maximum/minimum) within the feasible region.
STEP 4: the optimal solution is at a corner point (vertex) of the feasible region, as per fundamental theorem of LP.

Question 6

STEP OF SIMPLEX METHOD

Answer 6

STEP 1: converting inequalities into equalities by introducing slack variables.
STEP 2: solving the system equations iteratively to move towards the optimal solution by improving the objective function at each step.

Question 7

THE SIMPLEX METHOD INVOLVES:

Answer 7

Initial feasible solution
iterative improvement
optimality test

Question 8

Key Theorems in Linear Programming

Answer 8

• Fundamental Theorem of Linear Programming
• Optimality Condition

Question 9

Special Cases in LP

Answer 9

infeasibility
unbounded solutions
degeneracy

Question 10

Software for Solving LP Problems

Answer 10

MATLAB
EXCEL SOLVER
LINDO
GAMS (GENERAL ALGEBRAIC MODELING SYSTEM)
PYTHON (WITH LIBRARIES LIKE PULP AND SCIPY)

Question 11

Linear programming is a mathematical method for determining the best possible outcome in a mathematical model with _______ relationships.

Answer 11

linear

Question 12

Linear programming is often used in _______ problems where the objective is to maximize or minimize a linear function.

Answer 12

optimization

Question 13

In linear programming, _______ _______ are the variables that we need to solve for and represent the unknowns of the problem.

Answer 13

decision variables

Question 14

The _______ _______ is a linear function that needs to be maximized or minimized in a linear programming problem.

Answer 14

objective function

Question 15

To formulate a linear programming problem, the first step is to identify the _______ _______.

Answer 15

decision variables

Question 16

In a graphical solution method for two-variable LP problems, the _______ _______ is the area where all constraints are satisfied.

Answer 16

feasible region

Question 17

The optimal solution in a graphical method is found at a _______ point (vertex) of the feasible region.

Answer 17

corner

Question 18

For larger LP problems, the _______ _______ is commonly used to find the optimal solution iteratively.

Answer 18

simplex method

Question 19

n the Simplex Method, _______ variables are introduced to convert inequalities into equalities.

Answer 19

slack

Question 20

The _______ principle states that if the primal has an optimal solution, then the dual also has an optimal solution, and the value of the objective function is the same for both.

Answer 20

duality

Question 21

In duality, the primal problem involves _______ the objective function subject to constraints.

Answer 21

maximizing

Question 22

The Fundamental Theorem of Linear Programming states that if an LP problem has an optimal solution, it occurs at one of the _______ of the feasible region.

Answer 22

vertices

Question 23

If no solution satisfies all constraints in an LP problem, the problem is said to be _______.

Answer 23

infeasible

Question 24

An LP problem is considered _______ if the objective function can be improved indefinitely without violating any constraints.

Answer 24

unbounded

Question 25

_______ occurs when more than one basic feasible solution exists, which can lead to cycling in the Simplex method.

Answer 25

Degeneracy

Question 26

_______ _______ provides insight into how sensitive the optimal solution is to changes in the coefficients of the objective function

Answer 26

Sensitivity Analysis

Question 27

In sensitivity analysis, it is important to understand how much the _______ can change without affecting the optimal solution.

Answer 27

constraints

Question 28

One software tool available for solving large-scale LP problems is _______ _______.

Answer 28

Excel Solver (other valid answers: MATLAB, LINDO, GAMS)

Question 29

_______ is a widely used software library in Python for solving linear programming problems.

Answer 29

PuLP (other valid answer: SciPy)