Deck 2: Introduction to Management Science Models

Question 1

Minimization linear programming models may not involve " $\le$ " constraints.

Answer 1

False

Question 2

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.

Answer 2

True

Question 3

Binary linear programming allows for the possibility of mutually exclusive constraints.

Answer 3

True

Question 4

"Range of optimality" describes the impact of simultaneous changes in objective function values and right-hand-side values.

Question 5

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.

Question 6

A linear programming problem with all " $\le$ " functional constraints and nonnegative right hand side values will never be infeasible.

Question 7

If two extreme points are optimal, then so is every point on the line segment connecting the two extreme points.

Question 8

A non-binding constraint is always a redundant constraint.

Question 9

In the branch-and-bound technique for solving integer linear programming models with a maximization objective function:

A)Once a feasible solution is found, it is optimal.
B)If the solution generated at one stage is X₁ = 5.7, X₂ = 3 but gives an objective function value higher than that of the best integer solution found to date, X₂ must be either 0, 1, or 2 in the optimal solution.
C)If the solution generated at one stage is X₁ = 5.7, X₂ = 3 but gives an objective function value higher than that of the best integer solution found to date, a new linear programming problem is solved with two constraints added: X₁ $\le$ 6 and X₁ $\ge$ 5.
D)If the solution generated at one stage is X₁ = 5.7, X₂ = 3 but gives an objective function value higher than that of the best integer solution found to date, two new linear programs are solved, one with the constraint

Question 10

When compared with standard linear programming, integer linear programming typically has:

A)more feasible solution points to evaluate.
B)fewer feasible solution points to evaluate.
C)the same number of feasible solution points to evaluate.
D)more linear constraints.

Question 11

The complementary slackness principle states that either there is zero slack on a constraint or the reduced cost is zero.

Question 12

Compared with standard linear programming algorithms, those for treating integer linear programming problems usually are:

A)less complex due to fewer possible solutions.
B)less amenable to "what if" analysis.
C)always heuristic.
D)easier to formulate.

Question 13

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.

Question 14

Each decision variable must appear in at least two constraints.

Question 15

The difference between a boundary point and an extreme point is the number of constraints satisfied.

Question 16

In solving an integer linear programming problem, lifting the integer requirements and first solving the underlying standard linear programming problem:

A)never yields an optimal solution.
B)is not part of the branch-and-bound technique.
C)is not part of the cutting plane technique.
D)may yield an integer-valued solution.

Question 17

An extreme point is an optimal solution.

Question 18

An optimal solution must have no slack on at least one constraint.

Question 19

Which of the following is not a necessary linear programming assumption?

A)The decision variable values are discrete.
B)The parameters are specified with certainty.
C)Constant returns to scale in the linear constraints and the object function coefficients.
D)No interactions permitted between decision variables.

Question 20

Linear programming and integer linear programming both yield a great amount of sensitivity analysis.

Question 21

Banner Tools produces two styles of steel hammers with wooden handles.The first sells for $6 and consists of .5 pounds of steel; the second sells for $15 and consists of 1 pound of steel.Since steel costs the firm $4 per pound and the handle, labor, and packaging costs amount to $1 for either hammer, the profits coefficients are $6 - .5($4) - $1 = $3 for the smaller hammer and $15 - 1($4) - $1 = $10 for the larger hammer.Thus the objective function for this model is MAX 3X₁ + 10X₂.Given that the shadow price for steel is $2, which of the following statements is correct?

A)Banner should not buy more steel.
B)Banner should buy all the steel it can only if it can purchase it for less than $2 per pound.
C)Banner should buy at least as much as the "ALLOWABLE INCREASE", but only if it can be purchased for less than $2 per pound.
D)Banner should buy at least as much as the "ALLOWABLE INCREASE", but only if it can be purchase for less than $6 per pound.

Question 22

The effect of deleting a linear constraint from a linear programming model depends on whether or not that constraint:

A)is a " $\le$ " or a " $\ge$ " constraint.
B)had negative coefficients.
C)is redundant.
D)is binding.

Question 23

Compare the points that the Simplex method moves to with the points touched by the graphical method for solving a two variable linear program.

Question 24

A "non-binding" constraint is:

A)redundant.
B)not satisfied with an equality at the optimal solution.
C)one having zero slack or surplus
D)never a non-negativity variable constraint.

Question 25

Which statement is not true if a maximization problem has an unbounded solution?

A)A data entry error has been made or a limiting constraint has been omitted.
B)The objective function value goes to + $\infty$ .
C)The values of all decision variables go to + $\infty$ .
D)The feasible region is unbounded.

Question 26

Define the additivity assumption.

Question 27

What is a linear programming model?

Question 28

What is wrong with this linear programming model?

Question 29

The feasible region does not include:

A)interior points.
B)boundary points.
C)points at which at least one of the decision variables is zero.
D)points which violate at least one of the functional or non-negativity constraints.

Question 30

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?

A)Add a constraint of airport gate usage.
B)Change the objective to maximize profits.
C)Increase values for demand.
D)Drop the crew assignment constraint from the model.

Question 31

The functional constraints of a linear model with nonnegative variables are 3X₁ + 5X₂ $\le$ 16 and 4X₁ + X₂ $\le$ 10.Which of the following points could not be an optimal solution for the model?

A)X₁ = 2.5, X₂ = 0
B)X₁ = 0, X₂ = 3.2
C)X₁ = 1, X₂ = 2.25
D)X₁ = 2, X₂ = 2

Question 32

An over-constrained linear programming problem results in what type of solution?

A)Unbounded.
B)Degenerate.
C)Infeasible.
D)Sub-optimal.

Question 33

Excel Solver reports "Solver could not find a feasible solution." What is your best logical alternative?

A)Change the objective function.
B)Run Excel Solver again.
C)Relax a constraint.
D)Add a constraint.

Question 34

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 - $\infty$ to 50 respectively.If the objective function coefficients are changed to 14, 31, and 45, the optimal solution:

A)will not change.
B)may not change.
C)will definitely change.
D)may change.

Question 35

If the points (5, 5, 5) and (7, 9, 5) are both optimal solutions to a linear programming problem, what other point is also optimal?

A)(6,7,8)
B)(6,7,5)
C)(7,5,5)
D)(5,9,5)

Question 36

Squire Leathers produces two sizes of wallets from cowhide.The first requires 60 squire inches of cowhide and the second requires 100 square inches.The company has 1000 square feet of cowhide.Part of the model is:

A)60X₁ + 100X₂ $\ge$ 144,000
B)60X₁ + 100X₂ $\le$ 144,000
C)60X₁ + 100X₂ = 144,000
D)60X₁ $\le$ 144,000 and 100X₂ $\le$ 144,000

Question 37

Which of the following constraints is redundant?

A)X₁ + X₂ $\le$ 10
B)X₁ - X₂ $\le$ 10
C)X₁ + 3X₂ $\le$ 20
D)X₁, X₂ $\ge$ 0

Question 38

Why would you not use linear programming to solve a problem with a single decision variable linear objective function and multiple constraints?

Question 39

The principle of "complementary slackness" implies that:

A)if the reduced cost is not zero, than the value of the decision variable is zero.
B)if a decision variable is zero, then its reduced cost must be positive.
C)if a decision variable is zero, then its reduced cost must be non-zero.
D)if a decision variable is zero, then its reduced cost must be zero.

Question 40

In a model with two decision variables, the restriction 3X₁ + 2X₂ $\le$ 6 represents:

A)a straight line.
B)the region of infeasibility.
C)an extreme point.
D)a linear constraint.

Question 41

A farmer has three possible cereal grain crops to grow in the coming planting season: barley, oats, and wheat.He has 240 acres of arable land: 160 are considered high grade and the remainder is low grade.He has $42,000 in available capital and can hire virtually unlimited hours of field labor, locally, for $6 per hour.Relevant crop factors are:Formulate this as a linear programming model.

Question 42

The University of Iowa is experimenting with a blend of soil amendments to be used in an analysis of variance study of the response of tomatoes to various amounts of sunlight.To perform this study, all other elements must be controlled so that the only variable is the sunlight.The minimum requirements for calcium, phosphorous, and potassium are 125 pounds, 150 pounds, and 120 pounds respectively.The soil amendment mixture from Prairie Gold consist of 25% calcium, 25% phosphorous, 12½% potassium, and 37½% other ingredients.It costs $0.40 per ounce.The mixture from Grinell Grow is 20% calcium, 25% phosphorous, 25% potassium, and 30% other ingredients.It sells for $0.50 per ounce.Formulate a linear programming model that will allow the University of Iowa to conduct this experiment using a minimum cost blend of the two soil amendment mixtures.

Question 43

Explain the different interpretations of shadow costs when the objective function is based on sunk costs versus included costs.

Question 44

The Carolina Chemical Company (CCC) makes two products, soap and shampoo, both in liquid form.The same two raw material inputs, glycerin and potash, and the same production line are used in the manufacture of each.Each production day, one ton of glycerin and two tons of potash are available.Each gallon of soap requires 8 pounds of glycerin and 32 pounds of potash, while a gallon of shampoo uses 12 pounds of glycerin and 19 pounds of potash.
CCC's production day is 8 full hours (480 minutes).Its production line can turn out 25 gallons of soap, or 20 gallons of shampoo, per hour.Only one product can be made at a time, but the line is easily changed over at negligible cost from one item to the other.Forecasts indicate that demand for soap and shampoo far exceeds production capacity.The profit per gallon of soap is $30 and per gallon of shampoo is $36.
A.Formulate a linear programming model that will maximize CCC's daily profit.
B.Suppose that the production of soap also yields a byproduct called Blyx.The manufacturing process yields 0.3 units of Blyx for every unit of soap.Up to 15 gallons of Blyx can be sold per day for canine shampoo at a profit of $10 per gallon.CCC must dispose of any excess of Blyx at a removal cost of $4 per gallon.Using X3 = the number of gallons of Blyx produced up to 15 gallons, and X4 = the number of gallons of Blyx produced over 15 gallons, modify the linear programming model for this problem.

Question 45

Explain the SUMPRODUCT function of Excel.

Question 46

Office2000 produces expensive, quality 2-drawer and 4-drawer filing cabinets from solid oak for major corporations.A 2-drawer model utilizes 2 labor hours to produce and package for a net profit of $75.The 4-drawer model utilizes 3 labor hours to produce and package and nets a profit of $125.Each month Office2000 has 360 labor hours available and can obtain up to 200 2-drawer frames, 200 4-drawer frames, and 400 drawers from its oak supplier.

Question 47

When running Excel, what does "The Set Cell values do not converge" mean?

Question 48

What clue does Excel give to the possible existence of alternate optimal solutions?

Question 49

"Allowable Decrease" and "Allowable Increase" refer to what range?

Question 50

A small foundry has received an order for an iron alloy called Falloy which has the following specifications:
$\begin{array} { l l } \text { Content } &{ \text { Specification } } \\\hline\text { Nickel } & 5 \% \\\text { Manganese } & 4 \% \\\text { Copper } & 1 - 2 \% \\\text { Impurities } & { \text { no more than 3\% } } \\\text { Iron } & { \text { the balance } }\end{array}$
Falloy can be produced from the following inputs (balance to 100% consisting of iron).
$\begin{array} { l l } \text { Content } &{ \text { Specification } } \\\hline\text { Nickel } & 5 \% \\\text { Manganese } & 4 \% \\\text { Copper } & 1 - 2 \% \\\text { Impurities } & { \text { no more than 3\% } } \\\text { Iron } & { \text { the balance } }\end{array}$

Formulate the linear programming problem that will indicate the blending "recipe" that will minimize the cost of this alloy.

Question 51

Jungle Figures, Inc.produces two models of its stuffed giraffes, which it markets to high-end retail stores.The large giraffe requires 2 pounds of stuffing material and 6 minutes of machine time.The small giraffe requires 1 pound of stuffing material and 12 minutes of machine time since its tighter stitching pattern requires it to be stitched twice.There are 800 pounds of stuffing and 70 machine hours available each week.By adhering to a policy of not producing more than twice the number of large giraffes as small giraffes (this is a constraint), Jungle Figures has been able to sell all the giraffes it produces at $12 per large giraffe and $9 per small giraffe.

Question 52

The Finast Filter Company makes two types of filters for domestic air purifiers: the standard (nicknamed the Cleaner) for ordinary use, and the Scrubber for industrial-strength problems.The Cleaners bring in a profit of $0.50 each, while the Scrubbers command a profit of $1.25 each.Due to a higher piece-rate wage for the Scrubbers, the labor union was able to negotiate a requirement that each day the number of Scrubber filters produced will be at least as great as the number of Cleaners.The production line can turn out 25 Scrubbers per minute or 75 Cleaners per minute.Only one type of filter can be made at a time, but the production line is easily changed over at a negligible cost from one kind of filter production to the other.
FFC must produce at least 5,000 Cleaners today to fill a rush order.The production day is 8 full hours (480 minutes).
Formulate a linear programming model to determine how many of each filter should be produced today to maximize profit.

Question 53

Consider the following sensitivity report from Excel for the model in question 3, which has a profit of $5,400.

A.If Jungle Figures eliminates its policy of not producing more than twice the number of large giraffes as small giraffes, how will this affect the optimal solution?
B.Suppose 50 pounds of extra stuffing could be purchased for $200.Should Jungle do this? Would the optimal solution change?
C.Jungle is considering diverting 10 hours of machine time from the production of other products to produce giraffes.It figures it will lose $225 in gross profits from those products by this action.Should Jungle do this? Would the optimal solution change?
D.Suppose that demand is such that Jungle could effectively raise its prices to $15 on each giraffe (large or small).Would the optimal production schedule change?
E.Suppose demand for small giraffes waned, and Jungle cut its process so that its profit on small giraffes was reduced by 50% to $4.50.Would the optimal solution change?

Question 54

Ira Wax solved an integer linear programming problem by setting up a linear programming model without the integer constraints and rounding the solution.List the possible problems with this approach.

Question 55

The Hawaii Surfboard Company produces three different styles of surfboards (the Oahu, the Kauai, and the Maui) made from combinations of three different materials, A, B, and C.The Oahu model nets a $35 profit and requires 1 unit of A, 1 unit of B, and 1 unit of C.The Kauai model nets a $75 profit and requires 4 units of A and 1 unit of B.The Maui model nets a $100 profit and requires 1 unit of A, 4 units of B, and 1 unit of C.There are 200 units of A, 200 units of B, and 100 units of C available weekly.
A.Formulate a linear programming model to determine the optimal weekly profit.
B.How would the model change if Hawaii Surfboard wanted to produce an equal number of each model?

Question 56

How do you designate variables as integers when using Excel Solver?

Question 57

What is the major difference between a product mix problem and a diet problem?

Question 58

Spee-D Delivery Service wants to blend two gasolines (R and S) for use in its trucks.For the upcoming period, at least 10,000 gallons are required.Spee-D has a 20,000 gallon storage facility.Any required quantities of R and S gasoline may be purchased from a local refinery and mixed.R is 87 octane and costs $0.90 a gallon.S is 93 octane and costs $1.20 per gallon.Spee-D's fleet of 100 trucks requires an octane of no less than 89.Octane of the mixed output is linearly proportional to input octane.For example, equal parts of R and S would yield a 90 octane output.There is no gasoline loss in the blending process.
Formulate a linear programming model for this problem.Is a summation variable advisable here? Explain.