Applied Stat - Exam #3
Please refer to Aunt Anastasia's case in HW3 on folio. Aunt Anastasia is planning for next spring, and she is considering making only two products. Based on the results from the linear program, which two products would you recommend that she make?
baskets and rabbits
Multiple optimal solutions occur when constraints are parallel to each other. t/f
fasle
A slack
is the amount by which the left side of a "<=" constraint is smaller than the right side.
A "non-binding" constraint is
not satisfied with an equality at the optimal solution.
Non-negativity constraints
restrict the decision variables to positive values.
All model parameters are assumed to be known with certainty. t/f
true
If two extreme points are optimal, then so is every point on the line segment connecting the two extreme points. t/f
true
When using the graphical method, only one of the four quadrants of an xy-axis needs to be drawn. t/f
true
The shadow price for a constraint that expresses that the availability of wood is 3000 board-feet is $0.50, and the range of feasibility is between 2800 and 4000 board-feet. Which of the following is not correct?
If only 2900 board-feet of wood are available, the optimal solution will not change.
________ is used to analyze changes in model parameters.
Sensitivity analysis
Nike considers to build a factory at Millville or Greenfield, but not both. Alternatively, Nike may choose to build at neither location. The appropriate linear constraint to express this restriction using binary variables Y1 and Y2 is:
Y1 + Y2 <= 1
A plant manager is attempting to determine the production schedule of various products to maximize profit. Assume that a machine hour constraint is binding. If the original amount of machine hours available is 200 minutes., and the range of feasibility is from 130 minutes to 300 minutes, providing two additional machine hours will result in
a different product mix, different total profit.
The ________ property of linear programming models indicates that the rate of change or slope of the objective function or a constraint is constant.
constant to scale
A non-binding constraint is always a redundant constraint. t/f
false
Slack variables are only associated with maximization problems. t/f
false
Excel Solver reports "Solver could not find a feasible solution." What is your best logical alternative
Relax a constraint.
The difference between a boundary point and an extreme point is the number of constraints satisfied. t/f
false
When compared with standard linear programming, integer linear programming typically has:
fewer feasible solution points to evaluate.
A parameter is a numerical value in the objective function and constraints. t/f
true
Please refer to Aunt Anastasia's case in HW3 on folio. Aunt Anastasia feels that her prices are too low, particularly for her eggs. How much would her profit have to increase on the eggs before it is profitable for her to make and sell eggs?
$1.00
Cully Furniture buys two products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75,000 to invest in shelves this week, and the warehouse has 18,000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the maximum profit?
$45,000
The production manager for Beer etc. produces two kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. The manager can obtain at most 4800 oz of malt per week and at most 3200 oz of wheat per week, respectively. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. What is the optimal weekly profit?
$800
Hong Securities has $300,000 to invest in four stocks and three bonds. X1, X2, X3, and X4 denote the amounts invested in each of the stocks, and Y1, Y2, and Y3 equal the amounts invested in each of the three bonds. Which of the following shows that at least 40% of the investment in stocks must be in stock 1?
0.6X1 - 0.4X2 - 0.4X3 - 0.4X4 >= 0
Cully Turniture buys two products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75,000 to invest in shelves this week, and the warehouse has 18,000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the storage space constraint?
100 B + 90 M <= 18000
Cully Furniture buys two products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75,000 to invest in shelves this week, and the warehouse has 18,000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. Which of the following is not a feasible purchase combination?
100 big shelves and 100 medium shelves
The production manager for the Coory soft drink company is considering the production of two kinds of soft drinks: regular (R) and diet(D). Two of the limited resources are production time (8 hours = 480 minutes per day) and syrup (1 of the ingredients), limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the time constraint?
2R + 4D <= 480
The production manager for Beer etc. produces two kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. The manager can obtain at most 4800 oz of malt per week and at most 3200 oz of wheat per week, respectively. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. Which of the following is not a feasible solution?
300 L and 200 D
An intern sets up a linear program to optimize the use of paper products in the men's 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
Which of the following statements is not true?
An infeasible solution violates all constraints.
Billyboy Toys' toy balls, bats, and gloves net profits, excluding fixed costs, of $7, $8, and $13 respectively. The products require 2, 3, and 5 production hours each. Using current facilities, 1600 production hours are available for the production of these products each month. If Billyboy also leases a second, smaller production facility for $3000 per month, this will increase the availability of production hours for these products by 800. This situation can be modeled using a mixed integer model that includes the following:
An objective function of: MAX 7X1 + 8X2 + 13X3 - 3000Y1 Constraints including: 2X1 + 3X2 + 5X3 - 800Y1 <= 1600 Variable constraints including X1, X2, X3 >= 0, Y1 = 0 or 1
Please refer to Aunt Anastasia's case in HW3 on folio. Aunt Anastasia can obtain an additional 10 hours of kiln capacity free of charge from a friend. If she did this, how would her profits be affected?
Cannot tell from the information provided.
Dean Air uses a linear programming model to schedule flights, assign crews and meet passenger demand. The objective function is to minimize downtime. Constraints include a limit on pilot hours, meeting passenger demand, and scheduled downtime. Which of the following cannot be accomplished with sensitivity analysis?
Change the objective to maximize profits.
What is the initial step in the process of building linear models?
Determine decision variables.
The production manager for the Coory soft drink company is considering the production of two kinds of soft drinks: regular (R) and diet (D). Two of her limited resources are production time (8 hours = 480 minutes per day) and syrup (1 of the ingredients), limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the objective function?
MAX $3R + $2D
In a linear programming problem, a valid objective function can be represented as:
Max 3x + 3y + 1/3 z
Please refer to the Aunt Anastasia's case in HW3 on folio. Aunt Anastasia's available hours for paint and seal have fallen from 80 hours to 60 hours because of other commitments. How will this affect her profits?
Profits will not change
Please refer to the Aunt Anastasia's case in HW3 on folio. Suppose the charitable organization contacted Aunt Anastasia and told her that they had underestimated the amount of rabbits they needed. Instead of 25 rabbits, they need 35. How would this affect Aunt Anastasia's profits?
Profits would decrease by $5
The production manager for the Coory soft drink company is considering the production of two kinds of soft drinks: regular and diet. Two of her limited resources are production time (8 hours per day) and syrup (1 of the ingredients), limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What are the optimal daily production quantities of each product and the optimal daily profit?
R = 90, D = 75, Z = $420
Which of the following is not a necessary linear programming assumption?
The decision variable values are discrete.
The functional constraints of a linear model with nonnegative variables are 3X1 + 5X2 <= 16 and 4X1 + X2 <= 10. Which of the following points could not be a boundary point?
X1 = 1, X2 = 2.25
Nike may build a factory at Millville (Y1) or it may not. It may also build a regional warehouse at the same site (W1). But Nike will not build a warehouse without also building a factory. So, its choices are: (1) neither factory nor warehouse; (2) factory only; or (3) factory and warehouse. The appropriate linear constraint to express this is:
Y1 - W1 >= 0
In a model with two decision variables, the restriction 3X1+2X2 <= 6 represents
a linear constraint
A linear programming model consists of
constraints. an objective function. decision variables. *all of the above
The ________ property of linear programming models indicates that the decision variables cannot be restricted to integer values and can take on any fractional value.
continuity
For a maximization problem, assume that a constraint is binding. If the original amount of a resource is 4 lbs. and the range of feasibility (sensitivity range) for this constraint is from 3 lbs. to 6 lbs., increasing the amount of this resource by 1 lb. will result in the
different product mix, different total profit.
"Range of optimality" describes the impact of simultaneous changes in objective function values and right-hand-side values. t/f
false
A feasible solution violates at least one of the constraints. t/f
false
A linear programming model consists of only decision variables and constraints. t/f
false
A linear programming model has a constraint that reflects a budget restriction of $100,000. The range of feasibility for this amount, reflected on the sensitivity report, is $85,000 to $325,000. Thus if the budget restriction is changed to $90,000, the optimal solution will not change. t/f
false
A minimization model of a linear program contains only surplus variables. t/f
false
A variable is a value that is usually a coefficient of a parameter in an equation. t/f
false
An example of a decision variable in a linear programming problem is profit maximization. t/f
false
An extreme point is an optimal solution. t/f
false
Each decision variable must appear in at least two constraints. t/f
false
If the objective function is parallel to a constraint, the constraint is infeasible. t/f
false
In linear programming models , objective functions can only be maximized. t/f
false
It takes two pounds of steel and three pounds of copper to make a particular product. If there are 100 pounds of steel and 100 pounds of cooper available, one constraint will be 2X1 + 3X2 <= 200. t/f
false
Linear programming and integer linear programming both yield a great amount of sensitivity analysis. t/f
false
Minimization linear programming models may not involve "<=" constraints. t/f
false
Surplus variables are only associated with minimization problems. t/f
false
The complementary slackness principle states that either there is zero slack on a constraint or the reduced cost is zero. t/f
false
The first step of the management science process is to define the problem. t/f
false
The objective function coefficient for X1 is currently $18 and for X2 is $29, and the ranges of optimality for these coefficients are between $15 and $20 and between $25 and $35, respectively. If the objective function coefficients for X1 and X2 decline by $2 each, since both coefficients are still within their ranges of optimality, the optimal solution is guaranteed to remain the same. t/f
false
The optimal solution for a graphical linear programming problem is the corner point that is the farthest from the origin. t/f
false
There is exactly one optimal solution point to a linear program. t/f
false
Typically, finding a corner point for the feasible region involves solving a set of three simultaneous equations. t/f
false
When graphical method is used, we do not keep the scale of the unit on the grid paper the same. t/f
false
The principle of "complementary slackness" implies that
if the reduced cost is not zero, then the value of the decision variable is zero.
For a maximization problem, the shadow price measures the ________ in the value of the optimal solution, per unit increase for a given ________.
improvement, resource
Without satisfying the non-negativity constraint, a solution that satisfies all the other constraints of a linear programming problem is called
infeasible.
Please refer to the Aunt Anastasia's case in HW3 on folio. Which additional resources would you recommend that Aunt Anastasia try to obtain?
kiln
Compared with standard linear programming algorithms, those for treating integer linear programming problems usually are:
less amenable to :what if" analysis.
Decision variables
measure how much or how many items to produce, purchase, hire, etc.
Multiple optimal solutions can occur when the objective function is ________ a constraint line.
parallel to
Sensitivity analysis is the analysis of the effect of ________ changes on the ________.
parameter, optimal solution
The feasible region does not include:
points which violate at least one of the functional or non-negativity constraints.
A shadow price reflects which of the following in a maximization problem?
the marginal gain in the objective that would be realized by adding one unit of a resource
A constraint is a linear relationship representing a restriction on decision making. t/f
true
A linear programming problem with all ?<=? functional constraints and nonnegative right hand side values will never be infeasible. t/f
true
An optimal solution must have no slack on at least one constraint. t/f
true
Graphical solutions to linear programming problems have an infinite number of possible objective function lines. t/f
true
In the graphical approach, simultaneous equations may be used to solve for the optimal solution point. t/f
true
Linear programming is a model consisting of linear relationships representing a firm's decisions given an objective and resource constraints. t/f
true
Linear programming models are a subset of constrained optimization models that require the assumptions of continuity of the variables, certainty of the coefficients, additivity of terms, and proportionality of costs, profits, and the use of resources to the value of the decision variables. t/f
true
Linear programming models exhibit linearity among all constraint relationships and the objective function. t/f
true
Linear programming problems can be formulated both algebraically and on spreadsheets. t/f
true
Nimble Automotive uses linear programming to produce a monthly production schedule for their manufacturing plant. Although the number of cars built is obviously an integer, the fractional part of a non-integer decision variable could be considered ?work in progress? at the end of the month. t/f
true
One of the reasons we cannot use sensitivity analysis for an integer linear program is that the shadow prices do not produce linear effects. That is, although in a linear program the shadow price for a resource represents the marginal improvement for each added unit of that resource, in an integer linear program, we cannot assume that each added unit of a resource will produce the same marginal change. t/f
true
Parameters are known, constant values that are usually coefficients of variables in equations. t/f
true
The feasible solution area contains infinite solutions to the linear program. t/f
true
The objective function always consists of either maximizing or minimizing some value. t/f
true
The objective function is a linear relationship reflecting the objective of an operation. t/f
true
The parameters of a model are the numbers in the data cells of a spreadsheet. t/f
true
The terms in the objective function or constraints are additive. t/f
true
The values of decision variables are continuous or divisible. t/f
true
When specifying linear constraints, the modeler must take into account the unit specification of the decision variables so that the units represented by the left side of the constraints are consistent with the units represented by the right side of the constraints. t/f
true
You are currently paying $12 per hour for labor, and labor costs are included in the calculation of the objective function coefficients of a maximization problem. The shadow price for labor printed on the sensitivity analysis report is $8. It would be economically beneficial to you if you could secure extra labor for $15 per hour. t/f
true
The production manager for Beer etc. produces two kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. The manager can obtain at most 4800 oz of malt per week and at most 3200 oz of wheat per week, respectively. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. If the production manager decides to produce of 400 bottles of light beer and 0 bottles of dark beer, it will result in slack of
wheat only
The objective function coefficients for X1, X2, and X3 are 15, 32, and 48 respectively. Excel prints that their ranges of optimality are from 10 to 20, from 30 to 40, and from negative infinity to 50 respectively. If the objective function coefficients are changed to 14, 31, and 45, the optimal solution
will not change
