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Q:
NARRBEGIN: SA_87_90Sinclair Plastics operates two chemical plants which produce polyethylene; the Ohio Valley plant which produces 5000 tons per month and the Lakeview plant which can produce 7000 tons per month. Sinclair sells its polyethylene to three different GM auto plants, Grand Rapids (demand = 3000 tons per month), Blue Ridge (demand = 5000 tons per month), and Sunset (demand = 4000 tons per month). The costs of shipping between the respective plants is shown in the table below:Grand RapidsBlue RidgeSunsetOhio Valley5040100Lakeview755075NARRENDWhat is the objective function in this problem?
Q:
NARRBEGIN: SA_87_90Sinclair Plastics operates two chemical plants which produce polyethylene; the Ohio Valley plant which produces 5000 tons per month and the Lakeview plant which can produce 7000 tons per month. Sinclair sells its polyethylene to three different GM auto plants, Grand Rapids (demand = 3000 tons per month), Blue Ridge (demand = 5000 tons per month), and Sunset (demand = 4000 tons per month). The costs of shipping between the respective plants is shown in the table below: Grand RapidsBlue RidgeSunsetOhio Valley5040100Lakeview755075NARRENDWhat are the decision variables in this problem?
Q:
NARRBEGIN: SA_83_86A company produces two products. Each product can be produced on either of two machines. The time (in hours) required to produce each product on each machine is shown below:Machine 1Machine 2Product 154Product 285Each month, 600 hours of time are available on each machine, and that customers are willing to buy up to the quantities of each product at the prices that are shown below: DemandsPricesMonth 1Month 2Month 1Month 2Product 1120200$60$15Product 2150130$70$35The company's goal is to maximize the revenue obtained from selling units during the next two months.NARRENDFind an optimal solution to the problem, assuming that the company will not produce any units in either month that it cannot sell in that month. What is the production plan, and what is the total revenue?
Q:
NARRBEGIN: SA_83_86A company produces two products. Each product can be produced on either of two machines. The time (in hours) required to produce each product on each machine is shown below:Machine 1Machine 2Product 154Product 285Each month, 600 hours of time are available on each machine, and that customers are willing to buy up to the quantities of each product at the prices that are shown below: DemandsPricesMonth 1Month 2Month 1Month 2Product 1120200$60$15Product 2150130$70$35The company's goal is to maximize the revenue obtained from selling units during the next two months.NARRENDWhat are the constraints in this problem?
Q:
NARRBEGIN: SA_83_86A company produces two products. Each product can be produced on either of two machines. The time (in hours) required to produce each product on each machine is shown below: Machine 1Machine 2Product 154Product 285Each month, 600 hours of time are available on each machine, and that customers are willing to buy up to the quantities of each product at the prices that are shown below: DemandsPrices Month 1Month 2Month 1Month 2Product 1120200$60$15Product 2150130$70$35The company's goal is to maximize the revenue obtained from selling units during the next two months.NARRENDWhat is the objective function in this problem?
Q:
NARRBEGIN: SA_83_86A company produces two products. Each product can be produced on either of two machines. The time (in hours) required to produce each product on each machine is shown below: Machine 1Machine 2Product 154Product 285Each month, 600 hours of time are available on each machine, and that customers are willing to buy up to the quantities of each product at the prices that are shown below: DemandsPrices Month 1Month 2Month 1Month 2Product 1120200$60$15Product 2150130$70$35The company's goal is to maximize the revenue obtained from selling units during the next two months.NARRENDWhat are the decision variables in this problem?
Q:
NARRBEGIN: SA_76_82A marketing research professor is conducting a telephone survey and needs to contact at least 160 wives, 140 husbands, 110 single adult males, and 120 single adult females. It costs $2 to make a daytime call and $4 (because of higher labor costs) to make an evening call. The table shown below lists the expected results. For example, 10% of all daytime calls are answered by a single male, and 15% of all evening calls are answered by a single female. Because of a limited staff, at most half of all phone calls can be evening calls. Determine how to minimize the cost of completing the survey.PercentagesDaytimeEveningWife25%25%Husband15%30%Single male10%25%Single female15%15%None35%5%NARREND(A) What is the objective function in this problem?(B) What are the constraints in this problem? Write an algebraic expression for each.(C) Find an optimal solution to the problem using the formulation given in (A) and (B). What is the call plan, and what is the total cost?(D) Implement the model in (C) in Excel Solver and obtain an answer report. Which constraints are binding on the optimal solution?(E) Obtain a sensitivity report for the model in (D). If the professor could cut the cost of evening calls from $4 to $3, what would the new calling plan be?(F) Again using the sensitivity report obtained for (E), suppose the professor could get by with just 100 calls for single females. What would the call costs be in that case? Explain your answer.
Q:
NARRBEGIN: SA_76_82A marketing research professor is conducting a telephone survey and needs to contact at least 160 wives, 140 husbands, 110 single adult males, and 120 single adult females. It costs $2 to make a daytime call and $4 (because of higher labor costs) to make an evening call. The table shown below lists the expected results. For example, 10% of all daytime calls are answered by a single male, and 15% of all evening calls are answered by a single female. Because of a limited staff, at most half of all phone calls can be evening calls. Determine how to minimize the cost of completing the survey.PercentagesDaytimeEveningWife25%25%Husband15%30%Single male10%25%Single female15%15%None35%5%NARRENDWhat are the decision variables in this problem?
Q:
NARRBEGIN: SA_70_75A farmer in Egypt owns 50 acres of land. He is going to plant each acre with cotton or corn. Each acre planted with cotton yields $400 profit; each with corn yields $200 profit. The labor and fertilizer used for each acre are given in the table below. Resources available include 150 workers and 200 tons of fertilizer. CottonCornLabor (Workers)53Fertilizer (Tons)62NARREND(A) Formulate a linear programming model that will enable the farmer to determine the number of acres that should be planted cotton and/or corn in order to maximize his profit.(B) Find an optimal solution to the model in (A) and determine the maximum profit.(C) Implement the model in (A) in Excel Solver and obtain an answer report. Which constraints are binding on the optimal solution?(D) Obtain a sensitivity report for the model in (A). How much should the farmer be willing to pay for an additional worker?(E) Suppose the farmer hires 10 additional workers. Can you use the sensitivity analysis obtained for (D) to determine his expected profit? Would his planting plan change? Explain your answer.(F) Suppose the farmer now wants to hire 20 additional workers, instead of just 10. Can you use the sensitivity analysis obtained for (D) to determine his expected profit? Explain your answer.
Q:
If a solution to an LP problem satisfies all of the constraints, then is must be feasible.
Q:
If an LP model does have an unbounded solution, then we must have made a mistake " either we made an input error or we omitted one or more constraints.
Q:
Infeasibility refers to the situation in which there are no feasible solutions to the LP model
Q:
Unboundedness refers to the situation in which the LP model has been formulated in such a way that the objective function is unbounded " that is, it can be made as large (for maximization problems) or as small (for minimization problems) as we like.
Q:
It helps to ensure that Solver can find a solution to a linear programming problem if the model is well-scaled; that is, all of the numbers are of roughly the same magnitude.
Q:
The additivity property of LP models implies that the sum of the contributions from the various activities to a particular constraint equals the total contribution to that constraint.
Q:
When the proportionality property of LP models is violated, then we generally must use non-linear optimization.
Q:
The proportionality property of LP models means that if the level of any activity is multiplied by a constant factor, then the contribution of this activity to the objective function, or to any of the constraints in which the activity is involved, is multiplied by the same factor.
Q:
The divisibility property of LP models simply means that we allow only integer levels of the activities.
Q:
Proportionality, additivity, and divisibility are three important properties that LP models possess, which distinguish them from general mathematical programming models:.
Q:
Suppose the allowable increase and decrease for shadow price for a constraint are $25 (increase) and $10 (decrease). If the right hand side of that constraint were to increase by $10 the optimal value of the objective function would change.
Q:
Suppose the allowable increase and decrease for an objective coefficient of a decision variable that has a current value of $50 are $25 (increase) and $10 (decrease). If the coefficient were to change from $50 to $60, the optimal value of the objective function would not change.
Q:
Reduced costs indicate how much the objective coefficient of a decision variable that is currently 0 or at its upper bound must change before that the value of that variable changes.
Q:
Shadow prices are associated with nonbinding constraints, and show the change in the optimal objective function value when the right side of the constraint equation changes by one unit.
Q:
Nonbinding constraints will always have slack, which is the difference between the two sides of the inequality in the constraint equation.
Q:
It is often useful to perform sensitivity analysis to see how, or if, the optimal solution to a linear programming problem changes as we change one or more model inputs.
Q:
If a constraint has the equation , then the constraint line passes through the points (0,20) and (30,0):
Q:
If a constraint has the equation , then the slope of the constraint line is function line is -2:
Q:
If the objective function has the equation , then the y-intercept of the objective function line is 40:
Q:
If the objective function has the equation , then the slope of the objective function line is 2:
Q:
In determining the optimal solution to a linear programming problem graphically, if the objective is to maximize the objective, we pull the objective function line down until it contacts the feasible region.
Q:
The optimal solution to any linear programming model is a corner point of a polygon.
Q:
The feasible region in a graphical solution of a linear programming problem will appear as some type of polygon, with lines forming all sides.
Q:
It is instructive to look at a graphical solution procedure for LP models with three or more decision variables.
Q:
The set of all values of the changing cells that satisfy all constraints, not including the nonnegativity constraints, is called the feasible region.
Q:
When formulating a linear programming spreadsheet model, there is one target (objective) cell that contains the value of the objective function.
Q:
When formulating a linear programming spreadsheet model, there is a set of designated cells that play the role of the decision variables. These are called the objective cells.
Q:
When formulating a linear programming spreadsheet model, we specify the constraints in a Solver dialog box, since Excel does not show the constraints directly.
Q:
Linear programming problems can always be formulated algebraically, but not always on spreadsheet.
Q:
There are two primary ways to formulate a linear programming problem, the traditional algebraic way and in spreadsheets.
Q:
There are generally two steps in solving an optimization problem, model development and optimization.
Q:
There is often more than one objective in linear programming problems
Q:
In general, the complete solution of a linear programming problem involves three stages: formulating the model, invoking Solver to find the optimal solution, and performing sensitivity analysis.
Q:
All linear programming problems should have a unique solution, if they can be solved.
Q:
All optimization problems include decision variables, an objective function, and constraints.
Q:
One of the things that you can do with linear programming and a spreadsheet model is to develop a user interface to make it easier for someone who is not an expert in using linear programming. The output can be a report that explains the optimal policy in non-technical terms. The type of system being described is called a (n):
a. expert system
b. decision support system
c. linear programming support system
d. production planning system
Q:
Consider the following linear programming problem:
Minimize Subject to The above linear programming problem:
a. has only one optimal solution
b. has more than one optimal solution
c. exhibits infeasibility
d. exhibits unboundedness
Q:
Consider the following linear programming problem:
Maximize Subject to The above linear programming problem:
a. has only one optimal solution
b. has more than one optimal solution
c. exhibits infeasibility
d. exhibits unboundedness
Q:
Consider the following linear programming problem:
Maximize Subject to The above linear programming problem:
a. has only one optimal solution
b. has more than one optimal solution
c. exhibits infeasibility
d. exhibits unboundedness
Q:
Consider the following linear programming problem:
Maximize Subject to The above linear programming problem:
a. has only one optimal solution
b. has more than one optimal solution
c. exhibits infeasibility
d. exhibits unboundedness
Q:
In some cases, a linear programming problem can be formulated such that the objective can become infinitely large (for a maximization problem) or infinitely small (for a minimization problem). This type of problem is said to be:
a. infeasible
b. inconsistent
c. unbounded
d. redundant
Q:
When there is a problem with Solver being able to find a solution, many times it is an indication of a (n):
a. older version of Excel
b. nonlinear programming problem
c. problem that cannot be solved using linear programming
d. mistake in the formulation of the problem
Q:
The divisibility property of linear programming means that a solution can have both:
a. integer and noninteger levels of an activity
b. linear and nonlinear relationships
c. positive and negative values
d. revenue and cost information in the model
Q:
The additivity property of linear programming implies that the contribution of any decision variable to the objective is of/on the levels of the other decision variables.
a. dependent
b. independent
c. conditional
d. the sum
Q:
Linear programming models have three important properties. They are:
a. optimality, additivity and sensitivity
b. optimality, linearity and divisibility
c. divisibility, linearity and nonnegativity
d. proportionality, additivity and divisibility
Q:
In linear programming we can use the shadow price to calculate increases or decreases in:
a. binding constraints
b. nonbinding constraints
c. values of the decision variables
d. the value of the objective function
Q:
Related to sensitivity analysis in linear programming, when the profit increases with a unit increase in a resource, this change in profit is referred to as the:
a. add-in price
b. sensitivity price
c. shadow price
d. additional profit
Q:
In linear programming, sensitivity analysis involves examining how sensitive the optimal solution is to changes in:
a. profit of variables in model
b. cost of variables in model
c. resources available
d. All of these options
Q:
The equation of the line representing the constraint passes through the points:
a. b. c. d.
Q:
The equation of the line representing the constraint is:
a. b. c. d.
Q:
Suppose a company sells two different products, x and y, for net profits of $5 per unit and $10 per unit, respectively. The slope of the line representing the objective function is:
a. 0.5
b. -0.5
c. 2
d. -2
Q:
The optimal solution to any linear programming model is:
a. the maximum objective function line
b. the minimum objective function line
c. the corner point of a polygon
d. the maximum or minimum of a parabola
Q:
The feasible region in all linear programming problems is bounded by:
a. corner points
b. hyperplanes
c. an objective line
d. all of these options
Q:
When using the graphical solution method to solve linear programming problems, the set of points that satisfy all constraints is called the:
a. optimal region
b. feasible region
c. constrained region
d. logical region
Q:
A linear programming problem with only decision variable(s) can be solved by a graphical solution method.
a. 1
b. 2
c. 3
d. 4
Q:
Suppose a firm must at least meet minimum expected demands of 60 for product x and 80 of product y. An algebraic formulation of these constraints is:
a. b. c. d.
Q:
If a manufacturing process takes 3 hours per unit of x and 5 hours per unit of y and a maximum of 100 hours of manufacturing process time are available, then an algebraic formulation of this constraint is:
a. b. c. d.
Q:
Suppose a liquor store sells beer for a net profit of $1 per unit and wine for a net profit of $2 per unit. Let x equal the amount of beer sold and y equal the amount of wine sold. An algebraic formulation of the profit function is:
a. b. c. d.
Q:
The prototype linear programming problem is to select an optimal mix of products to produce to maximize profit. This type of problem is referred to as the:
a. product mix problem
b. production problem
c. product/process problem
d. product scheduling problem
Q:
In most cases in solving linear programming problems, we want the decision variables to be:
a. equal to zero
b. nonnegative
c. nonpositive
d. All of these options
Q:
The term nonnegativity refers to the condition where:
a. the objective function cannot be less that zero
b. the decision variables cannot be less than zero
c. the right hand side of the constraints cannot be greater then zero
d. the reduced cost cannot be less than zero
Q:
In using Excel to solve linear programming problems, the changing cells represent the:
a. value of the objective function
b. constraints
c. decision variables
d. total cost of the model
Q:
In using Excel to solve linear programming problems, the target cell represents the:
a. value of the objective function
b. constraints
c. decision variables
d. total cost of the model
Q:
The solution of a linear programming problem using Microsoft Excel typically involves the following three stages:
a. formulating the problem, invoking Solver, and sensitivity analysis
b. formulating the problem, graphing the problem, and sensitivity analysis
c. the changing cells, the target cells, and the constraints
d. the inputs, the changing cells, and the outputs
Q:
Every linear programming problem involves optimizing a:
a. linear regression model subject to several linear constraints
b. linear function subject to several linear constraints
c. linear function subject to several non-linear constraints
d. non-linear function subject to several linear constraints
Q:
The most important solution method for linear programming problems is known as the:
a. spreadsheet method
b. solution mix method
c. complex method
d. simplex method
Q:
Linear programming is a subset of a larger class of models called:
a. mathematical programming models
b. mathematical optimality models
c. linear regression models
d. linear simplex model
Q:
In an optimization model, there can only be one:
a. decision variable
b. constraint
c. objective function
d. shadow price
Q:
All optimization problems have:
a. an objective function and decision variables
b. an objective function and constraints
c. decision variables and constraints
d. an objective function, decision variables and constraints
Q:
NARRBEGIN: SA_119_122
The quarterly numbers of applications for home mortgage loans at a branch office of a large bank are recorded in the table below. NARREND
(A) Perform a runs test and compute a few autocorrelations to determine whether this time series is random.
(B) Obtain a time series chart. Which of the exponential smoothing models do you think should be used for forecasting based on this chart? Why?
(C) Use simple exponential smoothing to forecast these data, using no holdout period and requesting 4 quarters of future forecasts. Use the default smoothing constant of 0.10.
(D) Repeat (C), optimizing the smoothing constant. Does it make much of an improvement?
Q:
NARRBEGIN: SA_116_118
The table below contains 5 years of monthly data on sales (number of units sold) for a particular company, in addition to extra columns containing information needed to answer some of the questions. The company suspects that except for random noise, its sales are growing by a constant percentage each month and that they will continue to do so for at least the near future. NARREND
(A) Fit the appropriate regression model to the data. Report the resulting equation and state explicitly what it says about the percentage growth per month.
(B) What is the MAPE for the forecast model in (A)? What does it measure? Considering its magnitudes, does the model seem to be doing a good job?