Ops Final
For a linear programming problem, assume that a given resource has not been fully used. We can conclude that the shadow price associated with that constraint:
0
The reduced cost (shadow price) for a positive decision variable is __________.
0
In formulating a mixed integer programming problem, the constraint x1 + x2 ≤ 500y1 where y1 is a 0-1 variable and x1 and x2 are continuous variables, then x1 + x2 = 500 if y1 is __________.
1
LP Model Formulation Steps
1) Define the decision variables 2) Formulate the objective function 3) Formulate the constraints
Other types of sensitivity cases
1) add new constraints 2) add new decision variables 3) change constraint coefficients
If a maximization linear programming problem consist of all less-than-or-equal-to constraints with all positive coefficients and the objective function consists of all positive objective function coefficients, then rounding down the linear programming optimal solution values of the decision variables will __________ result in a feasible solution to the integer linear programming problem. A) always B) sometimes C) never
A
You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. A.) Assuming that Si is a binary variable, write the constraint for the first restriction. B.) Assuming that Si is a binary variable, write the constraint(s) for the second restriction. C.) Assuming that Si is a binary variable, write the constraint for the third restriction.
A) S1 + S3 +S7 ≤ 2 B) S2 +S5 ≤ 1, S4 +S5 ≤ 1 C) S1 + S2 + S3 +S4 + S5 + S6 + S7 ≥ 3
An intern sets up a linear program to optimize the use of paper products in the washroom. The system of equations he develops is: Max 2T + 3S + 4ST s.t 3T + 6S ≤ 40 10T + 10S ≤ 66 10T + 15S ≤ 99 His mentor studies the model, frowns, and admonishes the intern for violating which of the following properties of linear programming models?
Additivity
Standard form
All decision variables in the constraints must appear on the left hand side of the inequality (or equality) and all numerical values must be on the right-hand side
In a media selection problem, instead of having an objective of maximizing profit or minimizing cost, generally the objective is to maximize the __________.
Audience exposure
If the solution values of a linear program are rounded in order to obtain an integer solution, the solution is A) always optimal and feasible B) sometimes optimal and feasible C) always optimal D) always feasible E) never optimal and feasible
B
In a multi-period scheduling problem the production constraint usually takes the form of: A) beginning inventory + demand - production = ending inventory B) beginning inventory - demand + production = ending inventory C) beginning inventory - ending inventory + demand = production D) beginning inventory - production - ending inventory = demand E) beginning inventory + demand + production = ending inventory
B
__________ types of linear programming problems often result in fractional relations between variables which must be eliminated.
Blending
For a maximization integer linear programming problem, feasible solution is ensured by rounding __________ non-integer solution values if all of the constraints are less-than -or equal- to type. A) up and down B) up C) down D) up or down
C
For a resource constraint, either its slack value must be __________ or its shadow price must be __________. A) negative, negative B) negative, zero C) zero, zero D) zero, negative
C
Which of the following special cases does not require reformulation of the problem in order to obtain a solution? A) unboundedness B) infeasibility C) alternate optimality D) each one of these cases requires reformulation
C) alternate optimality
________ is not part of a Monte Carlo simulation. A.) Analyzing results B.) Analyzing a real problem C.) Finding an optimal solution D.) Evaluating the results E.) "What if" scenarios implementation
C.) Finding an optimal solution
If one location for a warehouse can be selected only if a specific location for a manufacturing facility is also selected, this decision can be represented by a __________ constraint.
Conditional
If a maximization linear programming problem consist of all less-than-or-equal-to constraints with all positive coefficients and the objective function consists of all positive objective function coefficients, then rounding down the linear programming optimal solution values of the decision variables will __________ result in a(n) __________ solution to the integer linear programming problem. A) always, optimal B) always, non-optimal C) never, non-optimal D) sometimes, optimal E) never, optimal
D
The theoretical limit on the number of constraints that can be handled by a linear programming problem is: A) 2. B) 3. C) 4. D) unlimited.
D
The type of linear program that compares services to indicate which one is less productive or inefficient is called
DEA
Divisibility
Decision variables can take on any fractional value and are therefore continuous as opposed to integer in nature
Infeasible Solutions
Every possible solution violates at least one constraint
A conditional constraint specifies the conditions under which variables are integers or real variables.
False
A feasible solution to an integer programming problem is ensured by rounding down integer solution values.
False
Data Envelopment Analysis indicates which type of service unit makes the highest profit.
False
For a profit maximization problem, if the allowable increase for a coefficient in the objective function is infinite, then profits are unbounded. True or False?
False
If we are solving a 0-1 integer programming problem, the constraint x1 + x2 + x3 ≤ 3 is a mutually exclusive constraint.
False
In a 0-1 integer programming problem involving a capital budgeting application (where xj = 1, if project j is selected, xj = 0, otherwise) the constraint x1 - x2 ≤ 0 implies that if project 2 is selected, project 1 can not be selected.
False
In a transportation problem, a demand constraint (the amount of product demanded at a given destination) is a less-than-or equal-to constraint (≤).
False
Linear programming model of a media selection problem is used to determine the relative value of each advertising media. True or False
False
Rounding down integer solution values ensures an infeasible solution to an integer linear programming problem.
False
Rounding non-integer solution values up to the nearest integer value will result in an infeasible solution to an integer linear programming problem.
False
Rounding non-integer solution values up to the nearest integer value will still result in a feasible solution.
False
Sensitivity ranges can be computed only for the coefficients of the objective function. True or False?
False
The "certainty" linear programming (LP) hypothesis (LP are deterministic models) is violated by integer programming. True or False?
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). True or False?
False
The optimal solution for a company that is able to produce two different products (x1 and x2) is x2 = 0 and x1 = 6. The best strategy for this company is to produce only x2. True or False?
False
The right hand side of constraints cannot be negative. True or False?
False
The standard form for the computer solution of a linear programming problem requires all variables to be to the right and all numerical values to be to the left of the inequality or equality sign. True or False.
False
Transportation problems can have solution values that are non-integer and must be rounded. True or False?
False
When using a linear programming model to solve the "diet" problem, the objective is generally to maximize nutritional content. True or False?
False
When using a linear programming model to solve the diet problem the objective is generally to maximize profit. True or False?
False
Increase in FC, _____ BE point
Increases
Increase in VC, _____ BE point
Increases
Continuous Probability Distributions
Integrate and set equal to r Solve for x
Increase in price, _____ BE point
Lowers
Productivity
Measure of efficiency— the amount of output produced compared to the amount of input required in production
The objective function of a diet problem is usually to __________ subject to nutritional requirements.
Minimize costs
model solution
Models solved using Management Sciences techniques
The sensitivity range for an objective function coefficient is the range of values over which the current __________________ remains the same.
Optimal Solution
The objective function 3x + 2y + 4xy violates the assumption of ________.
Proportionality
Properties of LP Problems
Proportionality Additivity Divisibility Certainty
Analogue simulation
Replaces a physical system with an analogous physical system that is easier to manipulate (example: conditions of weightlessness were simulated using rooms filled with water, wind tunnels that simulate the conditions of flight, and treadmills that simulate automobile tire wear in a laboratory instead of on the road, etc.)
Investment problems maximize __________.
Return on investments
Integer programming graphical solution
Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution - A feasible solution is ensured by rounding down non-integer solution values but may result in a less than optimal (sub-optimal) solution
Slack Variables
Should be in the form of an equation Added to a less then or equal to constraint Represents unused resources
Simulation
Simulation results will not equal analytical results unless enough trials have been conducted to reach steady state In simulation modeling, random numbers are generated by a mathematical process instead of a physical process (such as spinning a wheel) Random numbers are typically generated on the computer using a numerical technique and thus are not true random numbers but pseudorandom numbers One of the main attributes of simulation is its usefulness as a model for experimenting, called "what-if? Analysis"
Surplus Variables
Subtracted from a greater then or equal to constraint
The reduced cost (shadow price) for a positive decision variable is 0. True or False?
TRUE
Monte Carlo
Technique for selecting numbers randomly from a probability distribution for use in a trial (computer run) of a simulation model Values for a random variable are generated by sampling from a probability distribution The more periods simulated, the more accurate the results
Additivity
Terms in the objective function and constraints must be additive
What is Shadow Price?
The marginal value of one additional unit of resource: how much an organization is willing to pay for it Chg Z / Chg qi Also called dual variables
Multiple Optimal Solutions
The objective function is parallel to a constraint line
Certainty
The parameters values are assumed to be known with certainty (it is a deterministic model)
Proportionality
The rate of change (slope) of the objective function and constraints is constant
Changing the total amount of the constraint resource (right hand side)
The sensitivity range for a right-hand-side value is the range of values over which the quantity's value can change without changing the solution variable mix, including the slack/surplus variables The sensitivity range for a constraint quantity value is also the range over which the shadow price is valid To solve, find the x and y intercepts of each constraint and plug in the values into the opposite constraint to get qi
Changing objective function coefficients
The sensitivity range for an objective function coefficient is the range of values over which the current optimal solution point remains optimal The sensitivity range for the xi coefficient is designated as ci 1) Solve for the slope of the objective function 2) Solve for the slopes of the constraints 3) Set c1 and c2 equal to either of the constraint slopes and solve
Compared to blending and product mix problems, transportation problems are unique because
The solution values are always integers.
Graphical Solution of LP Models
They are limited to only to decision variables
What is the objective of the diet problem?
To minimize cost
A feasible solution to an integer programming problem is ensured by rounding down non- integer solution values.
True
A rounded-down integer solution can result in a less than optimal solution to an integer programming problem.
True
Blending problems usually require algebraic manipulation in order to write the LP in standard form
True
Determining the production quantities of different products manufactured by a company based on resource constraints is a product mix linear programming problem. True or False?
True
For a "Primal maximization problem" with Z equal to 2,570, its respective "Dual minimization problem " would also have a Z of 2,570. True or False?
True
Fractional relationships between variables are not permitted in the standard form of a linear program. True or False?
True
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 or False?
True
If the objective function slope is exactly the same as one of the constraints and this specific constraint is not redundant, we have a case of multiple optimal solutions. True or False?
True
If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a mutually exclusive constraint.
True
If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a conditional constraint.
True
In a balanced transportation model, supply equals demand such that all constraints can be treated as equalities. True or False?
True
In a media selection problem, instead of having an objective of maximizing profit or minimizing cost, generally the objective is to maximize the audience exposure. True or False?
True
In a media selection problem, maximization of audience exposure may not result in maximization of total profit. True or False?
True
In a problem involving capital budgeting applications, the 0-1 variables designate the acceptance or rejection of the different projects.
True
In a transportation problem, a demand constraint for a specific destination represents the amount of product demanded by a given destination (customer, retail outlet, store).
True
In a transportation problem, the supply constraint represents the maximum amount of product available for shipment or distribution at a given source (plant, warehouse, mill). True or False?
True
In an unbalanced transportation model, supply does not equal demand and supply constraints have ≤ signs. True or False?
True
One type of constraint in an integer program is a multiple choice constraint.
True
Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution to an integer programming problem.
True
Sensitivity analysis is a way to deal with uncertainty in linear programming models. True or False?
True
The 3 types of integer programming models are total, 0 - 1, and mixed.
True
The divisibility assumption is violated by integer programming.
True
Artificially created random numbers must have the following characteristics
Uniformly distributed Numerical technique for generating the numbers must be efficient The sequence of random numbers should reflect no pattern
For a minimization integer linear programming problem, a feasible solution is ensured by rounding ________ non-integer solution values if all of the constraints are the greater-than-or-equal-to type.
Up
Unbounded Solutions
Value of the objective function increases indefinitely
Computer mathematical simulation
When a system is replaced with a mathematical model that is analyzed with the computer. It offers a means of analyzing very complex systems that cannot be analyzed using other techniques.
If Xij = the production of product i in period j, write an expression to indicate that the limit on production of the company's 3 products in period 2 is equal to 400.
X12 + X22 + X32 ≤ 400
Linear Programming
a model that consists of linear relationships representing a firm's decision(s), given an objective function and resource constraints
Total Integer Model
all decision variables required to have integer solution variables x1, x2 >= 0 and integer
0-1 Integer Model
all decision variables required to have integer values of zero or one x1, x2, x3, x4 = 0 or 1
If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a __________ constraint.
corequisite
In a balanced transportation model, supply equals __________ .
demand
model construction
development of the functional mathematical relationship that describes the decision variables, objective function and constraints of the problem
Data Envelopment Analysis indicates the the relative _________ of a service unit compared with others.
efficiency or productivity
Observation
identification of a problem that exists in the system or organization
Objective function
linear relationship that reflects the objective of an operation
Model Constraints
linear relationship that represents a restriction on decision making
Decision variables
mathematical symbols that represent levels of activity
If exactly one investment is to be selected from a set of five investment options, then the constrain is often called a __________ constraint.
multiple-choice
If we are solving a 0-1 integer programming problem, the constraint x1 + x2 = 1 is a __________ constraint.
multiple-choice
Parameters
numerical values that are included in the objective functions and constraints
Definition of the problem
problem must be clearly and consistently defined showing its boundaries and interaction with the objectives of the organization
A croissant shop produces 2 products: bear claws (B) and almond filled croissants (C). Each bear claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS (tablespoons) of almond paste. An almond- filled croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for today's production run. The shop must produce at least 400 almond filled croissants due to customer demand. Bear claw profits are 20 cents each, and almond-filled croissant profits are 30 cents each. This represents what type of linear programming application?
product mix
Business competitiveness
refers to the ability of an organization to sell products in a market
For product mix problems, the constraints are usually associated with __________.
resources or time
Let: rj = regular production quantity for period j, oj =overtime production quantity in period j, ii = inventory quantity in period j, and di = demand quantity in period j Correct formulation of the demand constraint for a multi-period scheduling problem is:
rj + oj + i2 - i1 ≥ di
Mixed Integer Model
some of the decision variables (but not all) required to have integer values
Business competitiveness
the ability of an organization to sell products in a market
Break-even analysis
the aim is to determine the number of units of a product (volume) to sell or produce that will equate total revenue with total cost - zero profit
Contingency or mutually exclusive constraint
x1 + x2 <= 1
In an integer program, if we were choosing between two locations to build a facility, this would be written as: __________.
x1 + x2 =1
In an integer program, if building one facility required the construction of another type of facility, this would be written as: __________.
x1 = x2
Conditional constraint
x2 <= x1
Corequisite constraint
x2 = x1
