Ops Definitions Lectures 1-3
Transformation
is the conversion of inputs (resources) into (goods or services)
Objective Function
linear relationship that reflects the objective of an operation
Inputs
resources used in the process of production
Additivity
terms in the objective function and constraint equations must be additive
Proportionality
the rate of change (slope) of the objective function and constraint equations is constant
Consider the following linear programming problem: Max Z = $15x + $20y Subject to: 8x + 5y ≤ 40 0.4x + y ≥ 4 x, y ≥ 0 Considering that the values for x and y that will maximize revenue are respectively x = 0 and y = 8, what is the amount of slack/surplus associated with the first constraint - in the optimal point?
0
Proportionality, additivity, divisibility, and certainty
The properties of a linear programming problem are:
Which of the following combinations of constraints has no feasible region? x1 + x2 greater or equal than 15 and x1 - x2 less or equal than 10 x1 + x2 greater or equal than 5 and x1 greater or equal than 10 x1 + x2 greater or equal than 100 and x1 + x2 greater or equal than 50 x1 + x2 greater or equal than 10 and x1 greater or equal than 5
All of the above have a feasible region.
Standard Form
Changes the inequalities to equalities (=) and includes slack/surplus variables Ex. Max Z = 40x1 + 50x2 + 0s1 + 0s2 s.t. 1x1 + 2x2 + s1 = 40 (slack) 4x1 + 3x2 - s2 = 120 (surplus)
Shadow Price
Defined as the marginal value of one additional unit of resource: how much an organization is willing to pay for it ∆Z⁄∆qi
The break-even point is the volume that the profit is positive (greater than zero).
False
Linear Programming
a model that consists of linear relationships representing a firm's decision(s), given an objective and resource constraints
Model Construction
development of the functional mathematical relationship that describes the decision variables, objective function and constraints of the problem
Consider the following linear programming problem: Max Z = 4x1 + 2x2 Subject to: x1 less or equal than 4 x2 less or equal than 2 x1, x2 ≥ 0 The Z obtained from the best combination of x2 and x1 is:
20
Slack Variables
Added to a < constraint in order to convert it into an equation (=). They typically represent unused resources *Contributes nothing to the objective function (Z value)
___________________ refers to the ability of an organization to sell products in a market.
Business Competitiveness
Fixed cost includes: Answers: raw materials and resources staff and management salaries material handling and freight direct labor and packaging none of the above
Correct staff and management salaries
Simplex Method
Early LP used lengthy manual mathematical solution procedure called the
Answer the following statement with TRUE or FALSE: Sensitivity ranges can be computed only for the coefficients of the objective function.
False
Answer the following statement with TRUE or FALSE: the correct way to implement a sensitivity analysis for the second coefficient of the objective function is to vary not only the second coefficient but also the first one (at the same time).
False
If the objective function slope is exactly the same as one of the constraints and this specific constraint is redundant, we have a case of multiple optimal solutions.
False
Slack variables are only associated with maximization problems.
False
Surplus Variables
Subtracted from a > constraint in order to convert it to an equation (=). They represent excess above constraint requirement resources *Contributes nothing to the objective function (Z value)
Certainty
The parameters values are assumed to be known with certainty
Answer the following statement with TRUE or FALSE: The linear programming problem below is unbounded. Min Z = 5x1 + 3x2 s.t. 4x1 + 3x2 less or equal than 8 x1 less or equal than 4 x2 less or equal than 6
True
Answer the following statement with TRUE or FALSE: If one of the coefficients of the objective function is changed to a value outside of its respective sensitivity range (greater than the upper limit or lower than the lower limit), the optimal solution will be different than the one originally obtained before the change is implemented.
True
Answer the following statement with TRUE or FALSE: the optimal solution for a company that is able to produce two different products (x1 and x2) is x1 = 0 and x2 = 6. The best strategy for this company is to produce only x2.
True
The linear programming problem below is unbounded. Max Z = 5x1 + 3x2 s.t. 4x1 + 3x2 ≥ 8 x1 ≥ 4 x2 ≥ 6 x1, x2 ≥ 0 T/F?
True
Consider the following linear programming problem: Max Z = $200x1 + $100x2 Subject to: 8x1 + 5x2 ≤ 80 2x1 + x2 ≤ 100 x1, x2 ≥ 0 What is maximum Z and the value of x1 and x2 at the optimal solution?
Z = 2000; x1 = 10; and x2 = 0;
Observation
a problem that exists in the system or organization
Divisibility
decision variables can take on any fractional value and are therefore continuous as opposed to integer in nature
Sensitivity analysis
determines the effect on the optimal solution, for example, due to changes in coefficients of the objective function and/or constraints
If by processing the same amount of inputs used in the past a company is now capable to produce a(n) ________ amount of outputs, it means that an improvement of productivity was achieved. Please choose the option that best fits the empty space above.
greater
Decision Variables
mathematical symbols representing levels of activity of a firm
Productivity
measure of efficiency - the amount of output produced compared to the amount of input required in production
Model Solution
models solved using management sciences techniques
Parameters
numerical values that are included in the objective functions and constraints
When building a chair, a carpenter uses 2 pounds of wood and 3 ounces of glue. If the Carpenter has 3 pounds of wood and 6 ounces of glue, how many chairs will he/she be able to build?
one chair
The sensitivity range for an objective function coefficient is the range of values over which the current __________________ remains the same.
optimal solution
Definition of the Problem
problem must be clearly and consistently defined showing its boundaries and interaction with the objectives of the organization
Outputs
refer to the end result of an organisation's efforts — the service or product that is delivered or provided to the consumer
Business Competitiveness
refers to the ability of an organization to sell products in a market
The sensitivity range for right-hand-side value
the range of values over which the quantity's value can change without changing the solution variable mix, including the slack/surplus variables
The following linear programming problem has _________________ Max Z = $40x1 + $10x2 Subject to: 8x1 + 5x2 ≤ 80 2x1 + 1x2 ≥ 100 x1, x2 ≥ 0
No solution since it is infeasible
Answer the following statement with TRUE or FALSE: Sensitivity analysis is a way to deal with uncertainty in linear programming models.
True
The sensitivity range
_______________ for an objective function coefficient is the range of values over which the current optimal solution point remains optimal
Model Constraints
a linear relationship that represents a restriction on decision making
Model Implementation
actual use of the model or its solution
Productivity
Australian road freight company Linfox uses aerodynamic trucks and trailers to reduce fuel consumption. This is a case of generating higher