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