Exam 2: Introduction to Management Science Models

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?

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.

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

What is a linear programming model?

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

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.

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?

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

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

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.

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

Which of the following constraints is redundant?

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.

A "non-binding" constraint is:

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

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.

Each decision variable must appear in at least two constraints.

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.