problem solving with linear models

Our Goal: To predict Y from \(X _ { 1 } , X _ { 2 } , \text { and } X _ { 3 }\) .

This is a typical regression problem.

Upon successful completion of this lesson, you should be able to:

• Review of linear regression model focusing on prediction.
• Use least square estimation for linear regression.
• Apply model developed in training data to an independent test data.
• Context setting for more complex supervised prediction methods.

4.1 Linear Methods

The linear regression model:

\[ f(X)=\beta_{0} + \sum_{j=1}^{p}X_{j}\beta_{j}\]

This is just a linear combination of the measurements that are used to make predictions, plus a constant, (the intercept term). This is a simple approach. However, It might be the case that the regression function might be pretty close to a linear function, and hence the model is a good approximation.

What if the model is not true?

• It still might be a good approximation - the best we can do.
• Sometimes because of the lack of training data or smarter algorithms, this is the most we can estimate robustly from the data.

Comments on \(X_j\) :

• We assume that these are quantitative inputs [or dummy indicator variables representing levels of a qualitative input]
• We can also perform transformations of the quantitative inputs, e.g., log(•), √(•). In this case, this linear regression model is still a linear function in terms of the coefficients to be estimated. However, instead of using the original \(X_{j}\) , we have replaced them or augmented them with the transformed values. Regardless of the transformations performed on \(X _ { j } f ( x )\) is still a linear function of the unknown parameters.
• Some basic expansions: \(X _ { 2 } = X _ { 1 } ^ { 2 } , X _ { 3 } = X _ { 1 } ^ { 3 } , X _ { 4 } = X _ { 1 } \cdot X _ { 2 }\) .

Below is a geometric interpretation of a linear regression.

For instance, if we have two variables, \(X_{1}\) and \(X_{2}\) , and we predict Y by a linear combination of \(X_{1}\) and \(X_{2}\) , the predictor function corresponds to a plane (hyperplane) in the three-dimensional space of \(X_{1}\) , \(X_{2}\) , Y . Given a pair of \(X_{1}\) and \(X_{2}\) we could find the corresponding point on the plane to decide Y by drawing a perpendicular line to the hyperplane, starting from the point in the plane spanned by the two predictor variables.

For accurate prediction, hopefully, the data will lie close to this hyperplane, but they won’t lie exactly in the hyperplane (unless perfect prediction is achieved). In the plot above, the red points are the actual data points. They do not lie on the plane but are close to it.

How should we choose this hyperplane?

We choose a plane such that the total squared distance from the red points (real data points) to the corresponding predicted points in the plane is minimized. Graphically, if we add up the squares of the lengths of the line segments drawn from the red points to the hyperplane, the optimal hyperplane should yield the minimum sum of squared lengths.

The issue of finding the regression function \(E ( Y | X )\) is converted to estimating \(\beta _ { j } , j = 0,1 , \dots , p\) .

Remember in earlier discussions we talked about the trade-off between model complexity and accurate prediction on training data. In this case, we start with a linear model, which is relatively simple. The model complexity issue is taken care of by using a simple linear function. In basic linear regression, there is no explicit action taken to restrict model complexity. Although variable selection, which we cover in Lesson 5: Variable Selection, can be considered a way to control model complexity.

With the model complexity under check, the next thing we want to do is to have a predictor that fits the training data well.

Let the training data be:

\[\left\{ \left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) , \dots , \left( x _ { N } , y _ { N } \right) \right\} , \text { where } x _ { i } = \left( x _ { i 1 } , x _ { i 2 } , \ldots , x _ { i p } \right)\]

Denote \(\beta = \left( \beta _ { 0 } , \beta _ { 1 } , \ldots , \beta _ { p } \right) ^ { T }\) .

Without knowing the true distribution for X and Y , we cannot directly minimize the expected loss.

Instead, the expected loss \(E ( Y - f ( X ) ) ^ { 2 }\) is approximated by the empirical loss \(R S S ( \beta ) / N\) :

\[ \begin {align}RSS(\beta)&=\sum_{i=1}^{N}\left(y_i - f(x_i)\right)^2 &=\sum_{i=1}^{N}\left(y_i - \beta_0 -\sum_{j=1}^{p}x_{ij}\beta_{j}\right)^2 \\ \end {align} \]

This empirical loss is basically the accuracy you computed based on the training data. This is called the residual sum of squares, RSS .

The x ’s are known numbers from the training data.

Here is the input matrix X of dimension N × ( p +1):

\[\begin{pmatrix} 1 & x_{1,1} &x_{1,2} & ... &x_{1,p} \\ 1 & x_{2,1} & x_{2,2} & ... &x_{2,p} \\ ... & ... & ... & ... & ... \\ 1 & x_{N,1} &x_{N,2} &... & x_{N,p} \end{pmatrix}\]

Earlier we mentioned that our training data had N number of points. So, in the example where we were predicting the number of doctors, there were 101 metropolitan areas that were investigated. Therefore, N =101. Dimension p = 3 in this example. The input matrix is augmented with a column of 1’s (for the intercept term). So, above you see the first column contains all 1’s. Then if you look at every row, every row corresponds to one sample point and the dimensions go from one to p . Hence, the input matrix X is of dimension N × ( p +1).

Output vector y :

\[ y= \begin{pmatrix} y_{1}\\ y_{2}\\ ...\\ y_{N} \end{pmatrix} \]

Again, this is taken from the training data set.

The estimated \(\beta\) is \(\hat{\beta}\) and this is also put in a column vector, \(\left( \beta _ { 0 } , \beta _ { 1 } , \dots , \beta _ { p } \right)\) .

The fitted values (not the same as the true values) at the training inputs are

\[\hat{y}_{i}=\hat{\beta}_{0}+\sum_{j=1}^{p}x_{ij}\hat{\beta}_{j}\]

\[ \hat{y}= \begin{pmatrix} \hat{y}_{1}\\ \hat{y}_{2}\\ ...\\ \hat{y}_{N} \end{pmatrix} \]

For instance, if you are talking about sample i , the fitted value for sample i would be to take all the values of the x ’s for sample i , (denoted by \(x_{ij}\) ) and do a linear summation for all of these \(x_{ij}\) ’s with weights \(\hat{\beta}_{j}\) and the intercept term \(\hat{\beta}_{0}\) .

4.2 Point Estimate

• The least square estimation of \(\hat{\beta}\) is:

\[\hat{\beta} =(X^{T}X)^{-1}X^{T}y \]

• The fitted value vector is:

\[\hat{y} =X\hat{\beta}=X(X^{T}X)^{-1}X^{T}y \]

• Hat matrix:

\[H=X(X^{T}X)^{-1}X^{T} \]

Geometric Interpretation

Each column of X is a vector in an N -dimensional space (not the \(p + 1\) * dimensional feature vector space). Here, we take out columns in matrix X , and this is why they live in N*-dimensional space. Values for the same variable across all of the samples are put in a vector. I represent this input matrix as the matrix formed by the column vectors:

\[X = \left( X _ { 0 } , x _ { 1 } , \ldots , x _ { p } \right)\]

Here \(x_0\) is the column of 1’s for the intercept term. It turns out that the fitted output vector \(\hat{y}\) is a linear combination of the column vectors \(x _ { j } , j = 0,1 , \dots , p\) . Go back and look at the matrix and you will see this.

This means that \(\hat{y}\) lies in the subspace spanned by \(x _ { j } , j = 0,1 , \dots , p\) .

The dimension of the column vectors is N , the number of samples. Usually, the number of samples is much bigger than the dimension p . The true y can be any point in this N -dimensional space. What we want to find is an approximation constraint in the \(p+1\) dimensional space such that the distance between the true y and the approximation is minimized. It turns out that the residual sum of squares is equal to the square of the Euclidean distance between y and \(\hat{y}\) .

\[RSS(\hat{\beta})=\parallel y - \hat{y}\parallel^2 \]

For the optimal solution, \(y-\hat{y}\) has to be perpendicular to the subspace, i.e., \(\hat{y}\) is the projection of y on the subspace spanned by \(x _ { j } , j = 0,1 , \dots , p\) .

Geometrically speaking let’s look at a really simple example. Take a look at the diagram below. What we want to find is a \(\hat{y}\) that lies in the hyperplane defined or spanned by \(x _ {1}\) and \(x _ {2}\) . You would draw a perpendicular line from y to the plane to find \(\hat{y}\) . This comes from a basic geometric fact. In general, if you want to find some point in a subspace to represent some point in a higher dimensional space, the best you can do is to project that point to your subspace.

The difference between your approximation and the true vector has to be perpendicular to the subspace.

The geometric interpretation is very helpful for understanding coefficient shrinkage and subset selection (covered in Lesson 5 and 6).

4.3 Example Results

Let’s take a look at some results for our earlier example about the number of active physicians in a Standard Metropolitan Statistical Area (SMSA - data). If I do the optimization using the equations, I obtain these values below:

\[\hat{Y}_{i}= –143.89+0.341X_{i1}–0.019X_{i2}+0.254X_{i3} \]

\[RSS(\hat{\beta})=52,942,438 \]

Let’s take a look at some scatter plots. We plot one variable versus another. For instance, in the upper left-hand plot, we plot the pairs of \(x_{1}\) and y . These are two-dimensional plots, each variable plotted individually against any other variable.

STAT 501 on Linear Regression goes deeper into which scatter plots are more helpful than others. These can be indicative of potential problems that exist in your data. For instance, in the plots above you can see that \(x_{3}\) is almost a perfectly linear function of \(x_{1}\) . This might indicate that there might be some problems when you do the optimization. What happens is that if \(x_{3}\) is a perfectly linear function of \(x_{1}\) , then when you solve the linear equation to determine the \(β\) ’s, there is no unique solution. The scatter plots help to discover such potential problems.

In practice, because there is always measurement error, you rarely get a perfect linear relationship. However, you might get something very close. In this case, the matrix, \(X ^ { T } X\) , will be close to singular, causing large numerical errors in computation. Therefore, we would like to have predictor variables that are not so strongly correlated.

4.4 Theoretical Justification

If the linear model is true.

Here is some theoretical justification for why we do parameter estimation using least squares.

If the linear model is true, i.e., if the conditional expectation of Y given X indeed is a linear function of the X j ’s, and Y is the sum of that linear function and an independent Gaussian noise, we have the following properties for least squares estimation.

\[ E(Y|X)=\beta_0+\sum_{j=1}^{p}X_{j}\beta{j} \]

The least squares estimation of \(\beta\) is unbiased,

\[E(\hat{\beta}_{j}) =\beta_j, j=0,1, ... , p \]

To draw inferences about \(\beta\) , further assume: \(Y = E(Y | X) + \epsilon\) where \(\epsilon \sim N(0,\sigma^2)\) and is independent of X .

\(X_{ij}\) are regarded as fixed, \(Y_i\) are random due to \(\epsilon\) .

The estimation accuracy of \(\hat{\beta}\) , the variance of \(\hat{\beta}\) is given here:

\[Var(\hat{\beta})=(X^{T}X)^{-1}\sigma^2\]

You should see that the higher \(\sigma^2\) is, the variance of \(\hat{\beta}\) will be higher. This is very natural. Basically, if the noise level is high, you’re bound to have a large variance in your estimation. But then, of course, it also depends on \(X^T X\) . This is why in experimental design, methods are developed to choose X so that the variance tends to be small.

Note that \(\hat{\beta}\) is a vector and hence its variance is a covariance matrix of size ( p + 1) × ( p + 1). The covariance matrix not only tells the variance for every individual \(\beta_j\) , but also the covariance for any pair of \(\beta_j\) and \(\beta_k\) , \(j \ne k\) .

Gauss-Markov Theorem

This theorem says that the least squares estimator is the best linear unbiased estimator.

Assume that the linear model is true. For any linear combination of the parameters \(\beta_0 , \cdots ,beta_p\) you get a new parameter denoted by \(\theta = a^{T}\beta\) . Then \(a^{T}\hat{\beta}\) is just a weighted sum of \(\hat{\beta}_0, ..., \hat{\beta}_p\) and is an unbiased estimator since \(\hat{\beta}\) is unbiased.

We want to estimate \(θ\) and the least squares estimate of \(θ\) is:

\[ \begin {align} \hat{\theta} & = a^T\hat{\beta}\\ & = a^T(X^{T}X)^{-1}Xy \\ & \doteq \tilde{a}^{T}y, \\ \end{align} \]

which is linear in y . The Gauss-Markov theorem states that for any other linear unbiased estimator, \(c^Ty\) , the linear estimator obtained from the least squares estimation on \(\theta\) is guaranteed to have a smaller variance than \(c^Ty\) :

\[Var(\tilde{a}^{T}y) \le Var(c^{T}y).\]

Keep in mind that you’re only comparing with linear unbiased estimators. If the estimator is not linear, or is not unbiased, then it is possible to do better in terms of squared loss.

\(\beta_j\) , j = 0, 1, …, p are special cases of \(a^T\beta\) , where \(a^T\) only has one non-zero element that equals 1.

4.5 R Scripts

1. acquire data.

Diabetes data

The diabetes data set is taken from the UCI machine learning database on Kaggle: Pima Indians Diabetes Database

• 768 samples in the dataset
• 8 quantitative variables
• 2 classes; with or without signs of diabetes

Save the data into your working directory for this course as “diabetes.data.” Then load data into R as follows:

In RawData , the response variable is its last column; and the remaining columns are the predictor variables.

2. Fitting a Linear Model

In order to fit linear regression models in R, lm can be used for linear models, which are specified symbolically. A typical model takes the form of response~predictors where response is the (numeric) response vector and predictors is a series of predictor variables.

Take the full model and the base model (no predictors used) as examples:

For the full model, coefficients shows the least square estimation for \(\hat{\beta}\) and fitted.values are the fitted values for the response variable.

The results for the coefficients should be as follows:

The fitted values should start with 0.6517572852.

Source Code

Module 2: Linear Functions

Modeling with linear functions, learning outcomes.

• Build linear models from verbal descriptions.
• Build and solve systems of linear models.

Figure 1. (credit: EEK Photography/Flickr)

Emily is a college student who plans to spend a summer in Seattle. She has saved $3,500 for her trip and anticipates spending $400 each week on rent, food, and activities. How can we write a linear model to represent her situation? What would be the x -intercept, and what can she learn from it? To answer these and related questions, we can create a model using a linear function. Models such as this one can be extremely useful for analyzing relationships and making predictions based on those relationships. In this section, we will explore examples of linear function models.

Identifying Steps to Model and Solve Problems

When modeling scenarios with linear functions and solving problems involving quantities with a constant rate of change , we typically follow the same problem strategies that we would use for any type of function. Let’s briefly review them:

• Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.
• Carefully read the problem to identify important information. Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.
• Carefully read the problem to determine what we are trying to find, identify, solve, or interpret.
• Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.
• When needed, write a formula for the function.
• Solve or evaluate the function using the formula.
• Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.
• Clearly convey your result using appropriate units, and answer in full sentences when necessary.

Building Linear Models

Now let’s take a look at the student in Seattle. In her situation, there are two changing quantities: time and money. The amount of money she has remaining while on vacation depends on how long she stays. We can use this information to define our variables, including units.

• Output: M , money remaining, in dollars
• Input: t , time, in weeks

So, the amount of money remaining depends on the number of weeks: M ( t )

We can also identify the initial value and the rate of change.

• Initial Value: She saved $3,500, so $3,500 is the initial value for M .
• Rate of Change: She anticipates spending $400 each week, so –$400 per week is the rate of change, or slope.

Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. Also, because the slope is negative, the linear function is decreasing. This should make sense because she is spending money each week.

The rate of change is constant, so we can start with the linear model [latex]M\left(t\right)=mt+b[/latex]. Then we can substitute the intercept and slope provided.

To find the x -intercept, we set the output to zero, and solve for the input.

The x -intercept is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Emily will have no money left after 8.75 weeks.

When modeling any real-life scenario with functions, there is typically a limited domain over which that model will be valid—almost no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesn’t make sense to talk about input values less than zero. A negative input value could refer to a number of weeks before she saved $3,500, but the scenario discussed poses the question once she saved $3,500 because this is when her trip and subsequent spending starts. It is also likely that this model is not valid after the x -intercept, unless Emily will use a credit card and goes into debt. The domain represents the set of input values, so the reasonable domain for this function is [latex]0\le t\le 8.75[/latex].

In the above example, we were given a written description of the situation. We followed the steps of modeling a problem to analyze the information. However, the information provided may not always be the same. Sometimes we might be provided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the information we are given, and use it appropriately to build a linear model.

Using a Given Intercept to Build a Model

Some real-world problems provide the y -intercept, which is the constant or initial value. Once the y -intercept is known, the x -intercept can be calculated. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. Her loan balance is $1,000. She plans to pay $250 per month until her balance is $0. The y -intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is–$250 per month. We can then use the slope-intercept form and the given information to develop a linear model.

Now we can set the function equal to 0, and solve for x  to find the x -intercept.

The x -intercept is the number of months it takes her to reach a balance of $0. The   x -intercept is 4 months, so it will take Hannah four months to pay off her loan.

Using a Given Input and Output to Build a Model

Many real-world applications are not as direct as the ones we just considered. Instead they require us to identify some aspect of a linear function. We might sometimes instead be asked to evaluate the linear model at a given input or set the equation of the linear model equal to a specified output.

How To: Given a word problem that includes two pairs of input and output values, use the linear function to solve a problem.

• Identify the input and output values.
• Convert the data to two coordinate pairs.
• Find the slope.
• Write the linear model.
• Use the model to make a prediction by evaluating the function at a given x  value.
• Use the model to identify an x  value that results in a given y  value.
• Answer the question posed.

Example 1: Using a Linear Model to Investigate a Town’s Population

A town’s population has been growing linearly. In 2004 the population was 6,200. By 2009 the population had grown to 8,100. Assume this trend continues.

• Predict the population in 2013.
• Identify the year in which the population will reach 15,000.

The two changing quantities are the population size and time. While we could use the actual year value as the input quantity, doing so tends to lead to very cumbersome equations because the y -intercept would correspond to the year 0, more than 2000 years ago!

To make computation a little nicer, we will define our input as the number of years since 2004:

• Input: t , years since 2004
• Output: P ( t ), the town’s population

To predict the population in 2013 ( t  = 9), we would first need an equation for the population. Likewise, to find when the population would reach 15,000, we would need to solve for the input that would provide an output of 15,000. To write an equation, we need the initial value and the rate of change, or slope.

To determine the rate of change, we will use the change in output per change in input.

The problem gives us two input-output pairs. Converting them to match our defined variables, the year 2004 would correspond to [latex]t=0[/latex], giving the point [latex]\left(0,\text{6200}\right)[/latex]. Notice that through our clever choice of variable definition, we have “given” ourselves the y -intercept of the function. The year 2009 would correspond to [latex]t=\text{5}[/latex], giving the point [latex]\left(5,\text{8100}\right)[/latex].

The two coordinate pairs are [latex]\left(0,\text{6200}\right)[/latex] and [latex]\left(5,\text{8100}\right)[/latex]. Recall that we encountered examples in which we were provided two points earlier in the chapter. We can use these values to calculate the slope.

[latex]\begin{align} m&=\frac{8100 - 6200}{5 - 0} \\ &=\frac{1900}{5} \\ &=380\text{ people per year} \end{align}[/latex]

We already know the y -intercept of the line, so we can immediately write the equation:

[latex]P\left(t\right)=380t+6200[/latex]

To predict the population in 2013, we evaluate our function at t  = 9.

[latex]\begin{align}P\left(9\right)&=380\left(9\right)+6,200 \\ &=9,620\hfill \end{align}[/latex]

If the trend continues, our model predicts a population of 9,620 in 2013.

To find when the population will reach 15,000, we can set [latex]P\left(t\right)=15000[/latex] and solve for t .

[latex]\begin{gathered}15000=380t+6200 \\ 8800=380t \\ t\approx 23.158 \end{gathered}[/latex]

Our model predicts the population will reach 15,000 in a little more than 23 years after 2004, or somewhere around the year 2027.

A company sells doughnuts. They incur a fixed cost of $25,000 for rent, insurance, and other expenses. It costs $0.25 to produce each doughnut.

• Write a linear model to represent the cost C  of the company as a function of x , the number of doughnuts produced.
• Find and interpret the y -intercept.

[latex]C\left(x\right)=0.25x+25,000[/latex] The y -intercept is (0, 25,000). If the company does not produce a single doughnut, they still incur a cost of $25,000.

A city’s population has been growing linearly. In 2008, the population was 28,200. By 2012, the population was 36,800. Assume this trend continues.

• Predict the population in 2014.
• Identify the year in which the population will reach 54,000.

41,100; 2020

Using a Diagram to Model a Problem

It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output. The diagram should relate the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful.

Example 2: Using a Diagram to Model Distance Walked

Anna and Emanuel start at the same intersection. Anna walks east at 4 miles per hour while Emanuel walks south at 3 miles per hour. They are communicating with a two-way radio that has a range of 2 miles. How long after they start walking will they fall out of radio contact?

In essence, we can partially answer this question by saying they will fall out of radio contact when they are 2 miles apart, which leads us to ask a new question: “How long will it take them to be 2 miles apart?”

In this problem, our changing quantities are time and position, but ultimately we need to know how long will it take for them to be 2 miles apart. We can see that time will be our input variable, so we’ll define our input and output variables.

• Input: t , time in hours.
• Output: A ( t ), distance in miles, and E ( t ), distance in miles.

Because it is not obvious how to define our output variable, we’ll start by drawing a picture.

Initial Value: They both start at the same intersection so when [latex]t=0[/latex], the distance traveled by each person should also be 0. Thus the initial value for each is 0.

Rate of Change: Anna is walking 4 miles per hour and Emanuel is walking 3 miles per hour, which are both rates of change. The slope for A  is 4 and the slope for E  is 3.

Using those values, we can write formulas for the distance each person has walked.

[latex]\begin{align}&A\left(t\right)=4t\\ &E\left(t\right)=3t\end{align}[/latex]

For this problem, the distances from the starting point are important. To notate these, we can define a coordinate system, identifying the “starting point” at the intersection where they both started. Then we can use the variable, A , which we introduced above, to represent Anna’s position, and define it to be a measurement from the starting point in the eastward direction. Likewise, can use the variable, E , to represent Emanuel’s position, measured from the starting point in the southward direction. Note that in defining the coordinate system, we specified both the starting point of the measurement and the direction of measure.

We can then define a third variable, D , to be the measurement of the distance between Anna and Emanuel. Showing the variables on the diagram is often helpful.

Recall that we need to know how long it takes for D , the distance between them, to equal 2 miles. Notice that for any given input t , the outputs A ( t ), E ( t ), and D ( t ) represent distances.

This picture shows us that we can use the Pythagorean Theorem because we have drawn a right angle.

Using the Pythagorean Theorem, we get:

[latex]\begin{align}D{\left(t\right)}^{2}&=A{\left(t\right)}^{2}+E{\left(t\right)}^{2} \\ &={\left(4t\right)}^{2}+{\left(3t\right)}^{2} \\ &=16{t}^{2}+9{t}^{2} \\ &=25{t}^{2} \\ D\left(t\right)&=\pm \sqrt{25{t}^{2}} && \text{Solve for }D\left(t\right)\text{ using the square root} \\ &=\pm 5|t| \end{align}[/latex]

In this scenario we are considering only positive values of [latex]t[/latex], so our distance D ( t ) will always be positive. We can simplify this answer to D ( t ) = 5 t . This means that the distance between Anna and Emanuel is also a linear function. Because D  is a linear function, we can now answer the question of when the distance between them will reach 2 miles. We will set the output D ( t ) = 2 and solve for t .

[latex]\begin{align}D\left(t\right)&=2 \\ 5t&=2 \\ t&=\frac{2}{5}=0.4 \end{align}[/latex]

They will fall out of radio contact in 0.4 hours, or 24 minutes.

Should I draw diagrams when given information based on a geometric shape?

Yes. Sketch the figure and label the quantities and unknowns on the sketch.

Example 3: Using a Diagram to Model Distance between Cities

There is a straight road leading from the town of Westborough to Agritown 30 miles east and 10 miles north. Partway down this road, it junctions with a second road, perpendicular to the first, leading to the town of Eastborough. If the town of Eastborough is located 20 miles directly east of the town of Westborough, how far is the road junction from Westborough?

It might help here to draw a picture of the situation. It would then be helpful to introduce a coordinate system. While we could place the origin anywhere, placing it at Westborough seems convenient. This puts Agritown at coordinates (30, 10), and Eastborough at (20, 0).

Using this point along with the origin, we can find the slope of the line from Westborough to Agritown:

[latex]m=\frac{10 - 0}{30 - 0}=\frac{1}{3}[/latex]

The equation of the road from Westborough to Agritown would be

[latex]W\left(x\right)=\frac{1}{3}x[/latex]

From this, we can determine the perpendicular road to Eastborough will have slope [latex]m=-3[/latex]. Because the town of Eastborough is at the point (20, 0), we can find the equation:

[latex]\begin{align}E\left(x\right)&=-3x+b \\ 0&=-3\left(20\right)+b && \text{Substitute in (20, 0)} \\ b&=60 \\ E\left(x\right)&=-3x+60 \end{align}[/latex]

We can now find the coordinates of the junction of the roads by finding the intersection of these lines. Setting them equal,

[latex]\begin{align}\frac{1}{3}x&=-3x+60 \\ \frac{10}{3}x&=60 \\ 10x&=180 \\ x&=18 && \text{Substituting this back into }W\left(x\right) \\ y&=W\left(18\right) \\ &=\frac{1}{3}\left(18\right) \\ &=6 \end{align}[/latex]

The roads intersect at the point (18, 6). Using the distance formula, we can now find the distance from Westborough to the junction.

[latex]\begin{align}\text{distance}&=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}} \\ &=\sqrt{{\left(18 - 0\right)}^{2}+{\left(6 - 0\right)}^{2}} \\ &\approx 18.974\text{ miles} \end{align}[/latex]

Analysis of the Solution

One nice use of linear models is to take advantage of the fact that the graphs of these functions are lines. This means real-world applications discussing maps need linear functions to model the distances between reference points.

There is a straight road leading from the town of Timpson to Ashburn 60 miles east and 12 miles north. Partway down the road, it junctions with a second road, perpendicular to the first, leading to the town of Garrison. If the town of Garrison is located 22 miles directly east of the town of Timpson, how far is the road junction from Timpson?

21.15 miles

Building Systems of Linear Models

Real-world situations including two or more linear functions may be modeled with a system of linear equations . Remember, when solving a system of linear equations, we are looking for points the two lines have in common. Typically, there are three types of answers possible.

How To: Given a situation that represents a system of linear equations, write the system of equations and identify the solution.

• Identify the input and output of each linear model.
• Identify the slope and y -intercept of each linear model.
• Find the solution by setting the two linear functions equal to another and solving for x , or find the point of intersection on a graph.

Example 4: Building a System of Linear Models to Choose a Truck Rental Company

Jamal is choosing between two truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $20, then 59 cents a mile. The second, Move It Your Way, charges an up-front fee of $16, then 63 cents a mile 1 . When will Keep on Trucking, Inc. be the better choice for Jamal?

The two important quantities in this problem are the cost and the number of miles driven. Because we have two companies to consider, we will define two functions.

A linear function is of the form [latex]f\left(x\right)=mx+b[/latex]. Using the rates of change and initial charges, we can write the equations

[latex]\begin{gathered}K\left(d\right)=0.59d+20\\ M\left(d\right)=0.63d+16\end{gathered}[/latex]

Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Because all we have to make that decision from is the costs, we are looking for when Move It Your Way, will cost less, or when [latex]K\left(d\right)<M\left(d\right)[/latex]. The solution pathway will lead us to find the equations for the two functions, find the intersection, and then see where the [latex]K\left(d\right)[/latex] function is smaller.

These graphs are sketched in Figure 7, with K ( d ) in blue.

To find the intersection, we set the equations equal and solve:

[latex]\begin{align}K\left(d\right)&=M\left(d\right) \\ 0.59d+20&=0.63d+16 \\ 4&=0.04d \\ 100&=d \\ d&=100 \end{align}[/latex]

This tells us that the cost from the two companies will be the same if 100 miles are driven. Either by looking at the graph, or noting that [latex]K\left(d\right)[/latex] is growing at a slower rate, we can conclude that Keep on Trucking, Inc. will be the cheaper price when more than 100 miles are driven, that is [latex]d>100[/latex].

• Precalculus. Authored by : OpenStax College. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:1/Preface . License : CC BY: Attribution

Linear Functions Problems with Solutions

Linear functions are highly used throughout mathematics and are therefore important to understand. A set of problems involving linear functions , along with detailed solutions, are presented. The problems are designed with emphasis on the meaning of the slope and the y intercept.

Problem 1: f is a linear function. Values of x and f(x) are given in the table below; complete the table.

Problem 2: A family of linear functions is given by

Solution to Problem 2: a)

Problem 3: A high school had 1200 students enrolled in 2003 and 1500 students in 2006. If the student population P ; grows as a linear function of time t, where t is the number of years after 2003. a) How many students will be enrolled in the school in 2010? b) Find a linear function that relates the student population to the time t. Solution to Problem 3: a) The given information may be written as ordered pairs (t , P). The year 2003 correspond to t = 0 and the year 2006 corresponds to t = 3, hence the 2 ordered pairs (0, 1200) and (3, 1500) Since the population grows linearly with the time t, we use the two ordered pairs to find the slope m of the graph of P as follows m = (1500 - 1200) / (6 - 3) = 100 students / year The slope m = 100 means that the students population grows by 100 students every year. From 2003 to 2010 there are 7 years and the students population in 2010 will be P(2010) = P(2003) + 7 * 100 = 1200 + 700 = 1900 students. b) We know the slope and two points, we may use the point slope form to find an equation for the population P as a function of t as follows P - P1 = m (t - t1) P - 1200 = 100 (t - 0) P = 100 t + 1200

Problem 4: The graph shown below is that of the linear function that relates the value V (in $) of a car to its age t, where t is the number of years after 2000.

Problem 5: The cost of producing x tools by a company is given by

Problem 6: A 500-liter tank full of oil is being drained at the constant rate of 20 liters par minute. a) Write a linear function V for the number of liters in the tank after t minutes (assuming that the drainage started at t = 0). b) Find the V and the t intercepts and interpret them. e) How many liters are in the tank after 11 minutes and 45 seconds? Solution to Problem 6: After each minute the amount of oil in the tank deceases by 20 liters. After t minutes, the amount of oil in the tank decreases by 20*t liters. Hence if at the start there 500 liters, after t minute the amount V of oil left in the tank is given by V = 500 - 20 t b) To find the V intercept, set t = 0 in the equation V = 500 - 20 t. V = 500 liters : it is the amount of oil at the start of the drainage. To find the t intercept, set V = 0 in the equation V = 500 - 20 t and solve for t. 0 = 500 - 20 t t = 500 / 20 = 25 minutes : it is the total time it takes to drain the 500 liters of oil. c) Convert 11 minutes 45 seconds in decimal form. t = 11 minutes 45 seconds = 11.75 minutes Calculate V at t = 11.75 minutes. V(11.75) = 500 - 20*11.75 = 265 liters are in the tank after 11 minutes 45 seconds of drainage.

Problem 7: A 50-meter by 70-meter rectangular garden is surrounded by a walkway of constant width x meters.

Problem 8: A driver starts a journey with 25 gallons in the tank of his car. The car burns 5 gallons for every 100 miles. Assuming that the amount of gasoline in the tank decreases linearly, a) write a linear function that relates the number of gallons G left in the tank after a journey of x miles. b) What is the value and meaning of the slope of the graph of G? c) What is the value and meaning of the x intercept? Solution to Problem 8: a) If 5 gallons are burnt for 100 miles then (5 / 100) gallons are burnt for 1 mile. Hence for x miles, x * (5 / 100) gallons are burnt. G is then equal to the initial amount of gasoline decreased by the amount gasoline burnt by the car. Hence G = 25 - (5 / 100) x b) The slope of G is equal to 5 / 1000 and it represent the amount of gasoline burnt for a distance of 1 mile. c) To find the x intercept, we set G = 0 and solve for x. 25 - (5 / 100) x = 0 x = 500 miles : it is the distance x for which all 25 gallons of gasoline will be burnt.

Problem 9: A rectangular wire frame has one of its dimensions moving at the rate of 0.5 cm / second. Its width is constant and equal to 4 cm. If at t = 0 the length of the rectangle is 10 cm,

2.3 Modeling with Linear Functions

Learning objectives.

In this section, you will:

• Identify steps for modeling and solving.
• Build linear models from verbal descriptions.
• Build systems of linear models.

Emily is a college student who plans to spend a summer in Seattle. She has saved $3,500 for her trip and anticipates spending $400 each week on rent, food, and activities. How can we write a linear model to represent her situation? What would be the x -intercept, and what can she learn from it? To answer these and related questions, we can create a model using a linear function. Models such as this one can be extremely useful for analyzing relationships and making predictions based on those relationships. In this section, we will explore examples of linear function models.

Identifying Steps to Model and Solve Problems

When modeling scenarios with linear functions and solving problems involving quantities with a constant rate of change , we typically follow the same problem strategies that we would use for any type of function. Let’s briefly review them:

• Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.
• Carefully read the problem to identify important information. Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.
• Carefully read the problem to determine what we are trying to find, identify, solve, or interpret.
• Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.
• When needed, write a formula for the function.
• Solve or evaluate the function using the formula.
• Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.
• Clearly convey your result using appropriate units, and answer in full sentences when necessary.

Building Linear Models

Now let’s take a look at the student in Seattle. In her situation, there are two changing quantities: time and money. The amount of money she has remaining while on vacation depends on how long she stays. We can use this information to define our variables, including units.

• Output: M , M , money remaining, in dollars
• Input: t , t , time, in weeks

So, the amount of money remaining depends on the number of weeks: M ( t ) M ( t )

We can also identify the initial value and the rate of change.

• Initial Value: She saved $3,500, so $3,500 is the initial value for M . M .
• Rate of Change: She anticipates spending $400 each week, so –$400 per week is the rate of change, or slope.

Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. Also, because the slope is negative, the linear function is decreasing. This should make sense because she is spending money each week.

The rate of change is constant, so we can start with the linear model M ( t ) = m t + b . M ( t ) = m t + b . Then we can substitute the intercept and slope provided.

To find the x - x - intercept, we set the output to zero, and solve for the input.

The x - x - intercept is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Emily will have no money left after 8.75 weeks.

When modeling any real-life scenario with functions, there is typically a limited domain over which that model will be valid—almost no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesn’t make sense to talk about input values less than zero. A negative input value could refer to a number of weeks before she saved $3,500, but the scenario discussed poses the question once she saved $3,500 because this is when her trip and subsequent spending starts. It is also likely that this model is not valid after the x - x - intercept, unless Emily will use a credit card and goes into debt. The domain represents the set of input values, so the reasonable domain for this function is 0 ≤ t ≤ 8.75. 0 ≤ t ≤ 8.75.

In the above example, we were given a written description of the situation. We followed the steps of modeling a problem to analyze the information. However, the information provided may not always be the same. Sometimes we might be provided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the information we are given, and use it appropriately to build a linear model.

Using a Given Intercept to Build a Model

Some real-world problems provide the y - y - intercept, which is the constant or initial value. Once the y - y - intercept is known, the x - x - intercept can be calculated. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. Her loan balance is $1,000. She plans to pay $250 per month until her balance is $0. The y - y - intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is -$250 per month. We can then use the slope-intercept form and the given information to develop a linear model.

Now we can set the function equal to 0, and solve for x x to find the x - x - intercept.

The x - x - intercept is the number of months it takes her to reach a balance of $0. The x x -intercept is 4 months, so it will take Hannah four months to pay off her loan.

Using a Given Input and Output to Build a Model

Many real-world applications are not as direct as the ones we just considered. Instead they require us to identify some aspect of a linear function. We might sometimes instead be asked to evaluate the linear model at a given input or set the equation of the linear model equal to a specified output.

Given a word problem that includes two pairs of input and output values, use the linear function to solve a problem.

• Identify the input and output values.
• Convert the data to two coordinate pairs.
• Find the slope.
• Write the linear model.
• Use the model to make a prediction by evaluating the function at a given x - x - value.
• Use the model to identify an x - x - value that results in a given y - y - value.
• Answer the question posed.

Using a Linear Model to Investigate a Town’s Population

A town’s population has been growing linearly. In 2004 the population was 6,200. By 2009 the population had grown to 8,100. Assume this trend continues.

• ⓐ Predict the population in 2013.
• ⓑ Identify the year in which the population will reach 15,000.

The two changing quantities are the population size and time. While we could use the actual year value as the input quantity, doing so tends to lead to very cumbersome equations because the y - y - intercept would correspond to the year 0, more than 2000 years ago!

To make computation a little nicer, we will define our input as the number of years since 2004:

• Input: t , t , years since 2004
• Output: P ( t ) , P ( t ) , the town’s population

To predict the population in 2013 ( t = 9 ), ( t = 9 ), we would first need an equation for the population. Likewise, to find when the population would reach 15,000, we would need to solve for the input that would provide an output of 15,000. To write an equation, we need the initial value and the rate of change, or slope.

To determine the rate of change, we will use the change in output per change in input.

The problem gives us two input-output pairs. Converting them to match our defined variables, the year 2004 would correspond to t = 0 , t = 0 , giving the point ( 0 , 6200 ) . ( 0 , 6200 ) . Notice that through our clever choice of variable definition, we have “given” ourselves the y -intercept of the function. The year 2009 would correspond to t = 5, t = 5, giving the point ( 5 , 8100 ) . ( 5 , 8100 ) .

The two coordinate pairs are ( 0 , 6200 ) ( 0 , 6200 ) and ( 5 , 8100 ) . ( 5 , 8100 ) . Recall that we encountered examples in which we were provided two points earlier in the chapter. We can use these values to calculate the slope.

We already know the y -intercept of the line, so we can immediately write the equation:

To predict the population in 2013, we evaluate our function at t = 9. t = 9.

If the trend continues, our model predicts a population of 9,620 in 2013.

To find when the population will reach 15,000, we can set P ( t ) = 15000 P ( t ) = 15000 and solve for t . t .

Our model predicts the population will reach 15,000 in a little more than 23 years after 2004, or somewhere around the year 2027.

A company sells doughnuts. They incur a fixed cost of $25,000 for rent, insurance, and other expenses. It costs $0.25 to produce each doughnut.

• ⓐ Write a linear model to represent the cost C C of the company as a function of x , x , the number of doughnuts produced.
• ⓑ Find and interpret the y -intercept.

A city’s population has been growing linearly. In 2008, the population was 28,200. By 2012, the population was 36,800. Assume this trend continues.

• ⓐ Predict the population in 2014.
• ⓑ Identify the year in which the population will reach 54,000.

Using a Diagram to Model a Problem

It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output. The diagram should relate the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful.

Using a Diagram to Model Distance Walked

Anna and Emanuel start at the same intersection. Anna walks east at 4 miles per hour while Emanuel walks south at 3 miles per hour. They are communicating with a two-way radio that has a range of 2 miles. How long after they start walking will they fall out of radio contact?

In essence, we can partially answer this question by saying they will fall out of radio contact when they are 2 miles apart, which leads us to ask a new question:

“How long will it take them to be 2 miles apart?”

In this problem, our changing quantities are time and position, but ultimately we need to know how long will it take for them to be 2 miles apart. We can see that time will be our input variable, so we’ll define our input and output variables.

• Input: t , t , time in hours.
• Output: A ( t ) , A ( t ) , distance in miles, and E ( t ) , E ( t ) , distance in miles

Because it is not obvious how to define our output variable, we’ll start by drawing a picture such as Figure 2 .

Initial Value: They both start at the same intersection so when t = 0 , t = 0 , the distance traveled by each person should also be 0. Thus the initial value for each is 0.

Rate of Change: Anna is walking 4 miles per hour and Emanuel is walking 3 miles per hour, which are both rates of change. The slope for A A is 4 and the slope for E E is 3.

Using those values, we can write formulas for the distance each person has walked.

For this problem, the distances from the starting point are important. To notate these, we can define a coordinate system, identifying the “starting point” at the intersection where they both started. Then we can use the variable, A , A , which we introduced above, to represent Anna’s position, and define it to be a measurement from the starting point in the eastward direction. Likewise, can use the variable, E , E , to represent Emanuel’s position, measured from the starting point in the southward direction. Note that in defining the coordinate system, we specified both the starting point of the measurement and the direction of measure.

We can then define a third variable, D , D , to be the measurement of the distance between Anna and Emanuel. Showing the variables on the diagram is often helpful, as we can see from Figure 3 .

Recall that we need to know how long it takes for D , D , the distance between them, to equal 2 miles. Notice that for any given input t , t , the outputs A ( t ) , E ( t ) , A ( t ) , E ( t ) , and D ( t ) D ( t ) represent distances.

Figure 2 shows us that we can use the Pythagorean Theorem because we have drawn a right angle.

Using the Pythagorean Theorem, we get:

In this scenario we are considering only positive values of t , t , so our distance D ( t ) D ( t ) will always be positive. We can simplify this answer to D ( t ) = 5 t . D ( t ) = 5 t . This means that the distance between Anna and Emanuel is also a linear function. Because D D is a linear function, we can now answer the question of when the distance between them will reach 2 miles. We will set the output D ( t ) = 2 D ( t ) = 2 and solve for t . t .

They will fall out of radio contact in 0.4 hours, or 24 minutes.

Should I draw diagrams when given information based on a geometric shape?

Yes. Sketch the figure and label the quantities and unknowns on the sketch.

Using a Diagram to Model Distance between Cities

There is a straight road leading from the town of Westborough to Agritown 30 miles east and 10 miles north. Partway down this road, it junctions with a second road, perpendicular to the first, leading to the town of Eastborough. If the town of Eastborough is located 20 miles directly east of the town of Westborough, how far is the road junction from Westborough?

It might help here to draw a picture of the situation. See Figure 4 . It would then be helpful to introduce a coordinate system. While we could place the origin anywhere, placing it at Westborough seems convenient. This puts Agritown at coordinates ( 3 0 ,  1 0 ) , ( 3 0 ,  1 0 ) , and Eastborough at ( 2 0 , 0 ) . ( 2 0 , 0 ) .

Using this point along with the origin, we can find the slope of the line from Westborough to Agritown:

The equation of the road from Westborough to Agritown would be

From this, we can determine the perpendicular road to Eastborough will have slope m = – 3. m = – 3. Because the town of Eastborough is at the point (20, 0), we can find the equation:

We can now find the coordinates of the junction of the roads by finding the intersection of these lines. Setting them equal,

The roads intersect at the point (18, 6). Using the distance formula, we can now find the distance from Westborough to the junction.

One nice use of linear models is to take advantage of the fact that the graphs of these functions are lines. This means real-world applications discussing maps need linear functions to model the distances between reference points.

There is a straight road leading from the town of Timpson to Ashburn 60 miles east and 12 miles north. Partway down the road, it junctions with a second road, perpendicular to the first, leading to the town of Garrison. If the town of Garrison is located 22 miles directly east of the town of Timpson, how far is the road junction from Timpson?

Building Systems of Linear Models

Real-world situations including two or more linear functions may be modeled with a system of linear equations . Remember, when solving a system of linear equations, we are looking for points the two lines have in common. Typically, there are three types of answers possible, as shown in Figure 5 .

Given a situation that represents a system of linear equations, write the system of equations and identify the solution.

• Identify the input and output of each linear model.
• Identify the slope and y -intercept of each linear model.
• Find the solution by setting the two linear functions equal to one another and solving for x , x , or find the point of intersection on a graph.

Building a System of Linear Models to Choose a Truck Rental Company

Jamal is choosing between two truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $20, then 59 cents a mile. The second, Move It Your Way, charges an up-front fee of $16, then 63 cents a mile 4 . When will Keep on Trucking, Inc. be the better choice for Jamal?

The two important quantities in this problem are the cost and the number of miles driven. Because we have two companies to consider, we will define two functions.

A linear function is of the form f ( x ) = m x + b . f ( x ) = m x + b . Using the rates of change and initial charges, we can write the equations

Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Because all we have to make that decision from is the costs, we are looking for when Move It Your Way, will cost less, or when K ( d ) < M ( d ) . K ( d ) < M ( d ) . The solution pathway will lead us to find the equations for the two functions, find the intersection, and then see where the K ( d ) K ( d ) function is smaller.

These graphs are sketched in Figure 6 , with K ( d ) K ( d ) in blue.

To find the intersection, we set the equations equal and solve:

This tells us that the cost from the two companies will be the same if 100 miles are driven. Either by looking at the graph, or noting that K ( d ) K ( d ) is growing at a slower rate, we can conclude that Keep on Trucking, Inc. will be the cheaper price when more than 100 miles are driven, that is d > 100. d > 100.

Access this online resource for additional instruction and practice with linear function models.

• Interpreting a Linear Function

2.3 Section Exercises

Explain how to find the input variable in a word problem that uses a linear function.

Explain how to find the output variable in a word problem that uses a linear function.

Explain how to interpret the initial value in a word problem that uses a linear function.

Explain how to determine the slope in a word problem that uses a linear function.

Find the area of a parallelogram bounded by the y -axis, the line x = 3 , x = 3 , the line f ( x ) = 1 + 2 x , f ( x ) = 1 + 2 x , and the line parallel to f ( x ) f ( x ) passing through ( 2 ,  7 ) . ( 2 ,  7 ) .

Find the area of a triangle bounded by the x -axis, the line f ( x ) = 12 – 1 3 x , f ( x ) = 12 – 1 3 x , and the line perpendicular to f ( x ) f ( x ) that passes through the origin.

Find the area of a triangle bounded by the y -axis, the line f ( x ) = 9 – 6 7 x , f ( x ) = 9 – 6 7 x , and the line perpendicular to f ( x ) f ( x ) that passes through the origin.

Find the area of a parallelogram bounded by the x -axis, the line g ( x ) = 2 , g ( x ) = 2 , the line f ( x ) = 3 x , f ( x ) = 3 x , and the line parallel to f ( x ) f ( x ) passing through ( 6 , 1 ) . ( 6 , 1 ) .

For the following exercises, consider this scenario: A town’s population has been decreasing at a constant rate. In 2010 the population was 5,900. By 2012 the population had dropped 4,700. Assume this trend continues.

Predict the population in 2016.

Identify the year in which the population will reach 0.

For the following exercises, consider this scenario: A town’s population has been increased at a constant rate. In 2010 the population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues.

Identify the year in which the population will reach 75,000.

For the following exercises, consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.

Find the linear function that models the town’s population P P as a function of the year, t , t , where t t is the number of years since the model began.

Find a reasonable domain and range for the function P . P .

If the function P P is graphed, find and interpret the x - and y -intercepts.

If the function P P is graphed, find and interpret the slope of the function.

When will the output reached 100,000?

What is the output in the year 12 years from the onset of the model?

For the following exercises, consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year.

Find the linear function that models the baby’s weight W W as a function of the age of the baby, in months, t . t .

Find a reasonable domain and range for the function W W .

If the function W W is graphed, find and interpret the x - and y -intercepts.

If the function W is graphed, find and interpret the slope of the function.

When did the baby weight 10.4 pounds?

What is the output when the input is 6.2? Interpret your answer.

For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were afflicted.

Find the linear function that models the number of people inflicted with the common cold C C as a function of the year, t . t .

Find a reasonable domain and range for the function C . C .

If the function C C is graphed, find and interpret the x - and y -intercepts.

If the function C C is graphed, find and interpret the slope of the function.

When will the output reach 0?

In what year will the number of people be 9,700?

For the following exercises, use the graph in Figure 7 , which shows the profit, y, y, in thousands of dollars, of a company in a given year, t, t, where t t represents the number of years since 1980.

Find the linear function y , y , where y y depends on t , t , the number of years since 1980.

Find and interpret the y -intercept.

Find and interpret the x -intercept.

Find and interpret the slope.

For the following exercises, use the graph in Figure 8 , which shows the profit, y , y , in thousands of dollars, of a company in a given year, t , t , where t t represents the number of years since 1980.

For the following exercises, use the median home values in Mississippi and Hawaii (adjusted for inflation) shown in Table 2 . Assume that the house values are changing linearly.

In which state have home values increased at a higher rate?

If these trends were to continue, what would be the median home value in Mississippi in 2010?

If we assume the linear trend existed before 1950 and continues after 2000, the two states’ median house values will be (or were) equal in what year? (The answer might be absurd.)

For the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown in Table 3 . Assume that the house values are changing linearly.

If these trends were to continue, what would be the median home value in Indiana in 2010?

Real-World Applications

In 2004, a school population was 1,001. By 2008 the population had grown to 1,697. Assume the population is changing linearly.

• ⓐ How much did the population grow between the year 2004 and 2008?
• ⓑ How long did it take the population to grow from 1,001 students to 1,697 students?
• ⓒ What is the average population growth per year?
• ⓓ What was the population in the year 2000?
• ⓔ Find an equation for the population, P , P , of the school t years after 2000.
• ⓕ Using your equation, predict the population of the school in 2011.

In 2003, a town’s population was 1,431. By 2007 the population had grown to 2,134. Assume the population is changing linearly.

• ⓐ How much did the population grow between the year 2003 and 2007?
• ⓑ How long did it take the population to grow from 1,431 people to 2,134 people?
• ⓔ Find an equation for the population, P P of the town t t years after 2000.
• ⓕ Using your equation, predict the population of the town in 2014.

A phone company has a monthly cellular plan where a customer pays a flat monthly fee and then a certain amount of money per minute used on the phone. If a customer uses 410 minutes, the monthly cost will be $71.50. If the customer uses 720 minutes, the monthly cost will be $118.

• ⓐ Find a linear equation for the monthly cost of the cell plan as a function of x , the number of monthly minutes used.
• ⓑ Interpret the slope and y -intercept of the equation.
• ⓒ Use your equation to find the total monthly cost if 687 minutes are used.

A phone company has a monthly cellular data plan where a customer pays a flat monthly fee of $10 and then a certain amount of money per megabyte (MB) of data used on the phone. If a customer uses 20 MB, the monthly cost will be $11.20. If the customer uses 130 MB, the monthly cost will be $17.80.

• ⓐ Find a linear equation for the monthly cost of the data plan as a function of x x , the number of MB used.
• ⓒ Use your equation to find the total monthly cost if 250 MB are used.

In 1991, the moose population in a park was measured to be 4,360. By 1999, the population was measured again to be 5,880. Assume the population continues to change linearly.

• ⓐ Find a formula for the moose population, P since 1990.
• ⓑ What does your model predict the moose population to be in 2003?

In 2003, the owl population in a park was measured to be 340. By 2007, the population was measured again to be 285. The population changes linearly. Let the input be years since 1990.

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The Federal Helium Reserve held about 16 billion cubic feet of helium in 2010 and is being depleted by about 2.1 billion cubic feet each year.

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Suppose the world’s oil reserves in 2014 are 1,820 billion barrels. If, on average, the total reserves are decreasing by 25 billion barrels of oil each year:

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• ⓒ If the rate at which the reserves are decreasing is constant, when will the world’s oil reserves be depleted?

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When hired at a new job selling jewelry, you are given two pay options:

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• 4 Rates retrieved Aug 2, 2010 from http://www.budgettruck.com and http://www.uhaul.com/

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A MIP-heuristic approach for solving a bi-objective optimization model for integrated production planning of sugarcane and energy-cane

• Original Research
• Published: 02 September 2024

Cite this article

• Gilmar Tolentino 1   na1 ,
• Antônio Roberto Balbo   ORCID: orcid.org/0000-0002-4512-0140 1   na1 ,
• Sônia Cristina Poltroniere 1   na1 ,
• Angelo Aliano Filho   ORCID: orcid.org/0000-0002-5088-134X 2   na1 &
• Helenice de Oliveira Florentino   ORCID: orcid.org/0000-0003-2740-8826 3   na1  

This paper proposes a modeling and solution approach for the integrated planning of the planting and harvesting of sucrose cane and energy-cane considering multiple harvesters. An integer linear bi-objective optimization model is proposed, which seeks to find a trade-off between the maximization of the production volumes of sucrose and fiber and the minimization of the operational costs. The model considers the technical constraints of the mill, such as the milling capacity and meeting the monthly demand. A MIP-heuristic based on relax-and-fix and fix-and-optimize strategies with exact decomposition is appropriately proposed to determine approximations to Pareto optimal solutions to this problem. These approximations are used as incumbents for a branch-and-bound tree to generate potentially Pareto optimal solutions. The results reveal that the MIP-heuristic efficiently solves the problem for real and semi-random instances, generating approximate solutions with a reduced error and a reasonable computational effort. Moreover, the different solutions quantify the trade-off between cost and production volume, opening up the possibility of increasing sucrose and fiber content or decreasing the costs of solutions found. Thus, the proposed bi-objective approach, the solution technique and the different Pareto optimal solutions obtained can assist mill managers in making better decisions in sugarcane production.

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Agarwal, A. K. (2007). Biofuels (alcohols and biodiesel) applications as fuels for internal combustion engines. Progress in Energy and Combustion Science, 33 (3), 233–271.

Article   Google Scholar  

Akartunali, K., & Miller, A. J. (2009). A heuristic approach for big bucket multi-level production planning problems. European Journal of Operational Research, 193 (2), 396–411. https://doi.org/10.1016/j.ejor.2007.11.033

Aliano, A. F., Cantane, D. R., Isler, P. R., & Florentino, H. O. (2023). An integrated multi-objective mathematical model for sugarcane harvesting considering cumulative degree-days. Expert Systems with Applications, 232 , 120881.

Aliano, A. F., Melo, T., & Pato, M. V. (2021). A bi-objective mathematical model for integrated planning of sugarcane harvesting and transport operations. Computers & Operations Research, 134 , 105419.

Aliano, A. F., Moretti, A. C., Pato, M. V., & de Oliveira, W. A. (2021). An exact scalarization method with multiple reference points for bi-objective integer linear optimization problems. Annals of Operations Research, 296 (1), 35–69.

Aliano, F. A., de Oliveira, W. A., & Melo, T. (2022). Multi-objective optimization for integrated sugarcane cultivation and harvesting planning. European Journal of Operational Research, 309 (1), 330–344.

Beraldi, P., Ghiani, G., Grieco, A., & Guerriero, E. (2008). Rolling-horizon and fix-and-relax heuristics for the parallel machine lot-sizing and scheduling problem with sequence-dependent set-up costs. Computers & Operations Research, 35 (11), 3644–3656.

Bezanson, J., Edelman, A., Karpinski, S., & Shah, V. B. (2017). Julia: A fresh approach to numerical computing. SIAM Review, 59 (1), 65–98.

Bigaton, A., Danelon, A. F., Bressan, G., Silva, H. J. T., & Rosa, J. H. M. (2017). Previsão de custos do setor sucroenergético na região centro-sul do Brasil: safra 2017/18. Revista IPecege, 3 (3), 65–70.

Bruzzone, A. A. G., Anghinolfi, D., Paolucci, M., & Tonelli, F. (2012). Energy-aware scheduling for improving manufacturing process sustainability: A mathematical model for flexible flow shops. CIRP Annals - Manufacturing Technology, 61 (1), 459–462.

Calija, V., Higgins, A. J., Jackson, P. A., Bielig, L. M., & Coomans, D. (2001). An operations research approach to the problem of the sugarcane selection. Annals of Operations Research, 108 (1–4), 123–142.

Cárdenas-Barrón, L. E., Melo, R. A., & Santos, M. C. (2021). Extended formulation and valid inequalities for the multi-item inventory lot-sizing problem with supplier selection. Computers and Operations Research, 130 , 105234.

Cheavegatti-Gianotto, A., de Abreu, H. M. C., Arruda, P., Bespalhok Filho, J. C., Burnquist, W. L., Creste, S., di Ciero, L., Ferro, J. A., et al. (2011). Sugarcane ( saccharum x officinarum ): A reference study for the regulation of genetically modified cultivars in brazil. Tropical Plant Biology, 4 , 62–89.

Chu, S., & Majumdar, A. (2012). Opportunities and challenges for a sustainable energy future. Nature, 488 (7411), 294–303.

Cunha, J. O., Mateus, G. R., & Melo, R. A. (2022). A hybrid heuristic for capacitated three-level lot-sizing and replenishment problems with a distribution structure. Computers and Industrial Engineering, 173 , 108698. https://doi.org/10.1016/j.cie.2022.108698

Danna, E., Rothberg, E., & Pape, C. L. (2005). Exploring relaxation induced neighborhoods to improve MIP solutions. Mathematical Programming, 102 , 71–90.

Dunning, I., Huchette, J., & Lubin, M. (2017). Jump: A modeling language for mathematical optimization. SIAM Review, 59 (2), 295–320.

Ehrgott, M., & Wiecek, M. M. (2005). Multiobjective programming. Multiple Criteria Decision Analysis: State of the Art Surveys, 78 , 667–708.

Google Scholar  

Etemadnia, H., Goetz, S. J., Canning, P., & Tavallali, M. S. (2015). Optimal wholesale facilities location within the fruit and vegetables supply chain with bimodal transportation options: An lp-mip heuristic approach. European Journal of Operational Research, 244 (2), 648–661.

Farias, M. E. A., Martins, M. F., & Cândido, G. A. (2021). Agenda 2030 e energias renováveis: Sinergias e desafios para alcance do desenvolvimento sustentável. Research, Society and Development, 10 (17), e13101723867.

Ferreira, D., Morabito, R., & Rangel, S. (2009). Solution approaches for the soft drink integrated production lot sizing and scheduling problem. European Journal of Operational Research, 196 (2), 697–706. https://doi.org/10.1016/j.ejor.2008.03.035

Fischetti, M., & Lodi, A. (2003). Local branching. Mathematical Programming, 98 , 23–47.

Florentino, H. O., Irawan, C., Aliano, F. A., Jones, D. F., Cantane, D. R., & Nervis, J. J. (2018). A multiple objective methodology for sugarcane harvest management with varying maturation periods. Annals of Operations Research, 267 (1), 153–177.

Florentino, H. O., Jones, D. F., Irawan, C., Ouelhadj, D., Khosravi, B., & Cantane, D. R. (2020). An optimization model for combined selecting, planting and harvesting sugarcane varieties. Annals of Operations Research, 314 , 451–469.

Florentino, H. O., & Pato, M. V. (2014). A bi-objective genetic approach for the selection of sugarcane varieties to comply with environmental and economic requirements. Journal of the Operational Research Society, 65 (6), 842–854.

García-Segura, T., Penadés-Plà, V., & Yepes, V. (2018). Sustainable bridge design by metamodel-assisted multi-objective optimization and decision-making under uncertainty. Journal of Cleaner Production, 202 , 904–915.

Giagkiozis, I., & Fleming, P. J. (2015). Methods for multi-objective optimization: An analysis. Information Sciences, 293 , 338–350.

Gurobi Optimization, LLC: Gurobi Optimizer Reference Manual (2022). https://www.gurobi.com .

Helber, S., & Sahling, F. (2010). A fix-and-optimize approach for the multi-level capacitated lot sizing problem. International Journal of Production Economics, 123 (2), 247–256.

Higgins, A. (2006). Scheduling of road vehicles in sugarcane transport: A case study at an Australian sugar mill. European Journal of Operational Research, 170 (3), 987–1000.

Higgins, A., Antony, G., Sandell, G., Davies, I., Prestwidge, D., & Andrew, B. (2004). A framework for integrating a complex harvesting and transport system for sugar production. Agricultural Systems, 82 (2), 99–115.

Higgins, A. J., & Muchow, R. C. (2003). Assessing the potential benefits of alternative cane supply arrangements in the Australian sugar industry. Agricultural Systems, 76 (2), 623–638.

Higgins, A. J., & Postma, S. (2004). Australian sugar mills optimise siding rosters to increase profitability. Annals of Operations Research, 128 (1–4), 235–249.

James, R. J. W., & Almada-Lobo, B. (2011). Single and parallel machine capacitated lotsizing and scheduling: New iterative MIP-based neighborhood search heuristics. Computers and Operations Research, 38 (12), 1816–1825. https://doi.org/10.1016/j.cor.2011.02.005

Jena, S. D., & Poggi, M. (2013). Harvest planning in the Brazilian sugar cane industry via mixed integer programming. European Journal of Operational Research, 230 (2), 374–384.

Junqueira, R., & Morabito, R. (2019). Modeling and solving a sugarcane harvest front scheduling problem. International Journal of Production Economics, 213 , 150–160. https://doi.org/10.1016/j.ijpe.2019.03.009

Kaab, A., Sharifi, M., Mobli, H., Nabavi-Pelesaraei, A., & Chau, K. (2019). Use of optimization techniques for energy use efficiency and environmental life cycle assessment modification in sugarcane production. Energy, 181 , 1298–1320.

Kaewtrakulpong, K., Takigawa, T., Koike, M., Hasegawa, H., & Bahalayodhin, B. (2008). Truck allocation planning for cost reduction of mechanical sugarcane harvesting in Thailand an application of multi-objective optimization. Journal of the Japanese Society of Agricultural Machinery, 70 (2), 62–71.

Lamsal, K., Jones, P. C., & Thomas, B. W. (2017). Sugarcane harvest logistics in Brazil. Transportation Science, 51 (2), 771–789.

Liebman, M., & Dyck, E. (1993). Crop rotation and intercropping strategies for weed management. Ecological Applications, 3 (1), 92–122.

Lima, C., Balbo, A. R., Homem, T. P. D., & Florentino, H. O. (2017). A hybrid approach combining interior-point and branch-and-bound methods applied to the problem of sugar cane waste. Journal of the Operational Research Society, 68 (2), 147–164.

Matsuoka, S. & Rubio, L. C. S. (2019). Energy cane: A sound alternative of a bioenergy crop for tropics and subtropics. In Sugarcane Biofuels , pp. 39–66. Springer.

Matsuoka, S., dos Santos, E. G. D. & Tomazela, A. L. (2017). Free Fiber Level Drives Resilience and Hybrid Vigor in Energy Cane. LAP LAMBERT Academic Publishing.

Matsuoka, S., Rubio, L., Tomazela, A., & Santos, E. (2016). A evoluçao do proálcool. AgroANALYSIS, 36 (1), 29–30.

Miettinen, K. (1999). Nonlinear Multiobjective Optimization, vol. 12. Springer.

Milan, E. L., Fernandez, S. M., & Aragones, L. M. P. (2006). Sugar cane transportation in Cuba, a case study. European Journal of Operational Research, 174 (1), 374–386.

Muchow, R. C., Higgins, A. J., Rudd, A. V., & Ford, A. W. (1998). Optimising harvest date in sugar production: A case study for the Mossman mill region in Australia: II. sensitivity to crop age and crop class distribution. Field Crops Research, 57 (3), 243–251.

Nervis, J. J. (2015). Simulação para a otimização da colheita da cana-de-açúcar.

Nikulin, Y., Miettinen, K., & Mäkelä, M. M. (2012). A new achievement scalarizing function based on parameterization in multiobjective optimization. OR Spectrum, 34 (1), 69–87.

Pathumnakul, S., & Nakrachata-Amon, T. (2015). The applications of operations research in harvest planning: A case study of the sugarcane industry in Thailand. Journal of Japan Industrial Management Association, 65 (4E), 328–333.

Poltroniere, S. C., Aliano, F. A., Caversan, A. S., Balbo, A. R., & Florentino, H. O. (2021). Integrated planning for planting and harvesting sugarcane and energy-cane for the production of sucrose and energy. Computers and Electronics in Agriculture . https://doi.org/10.1016/j.compag.2020.105956

Pongpat, P., Mahmood, A., Ghani, H. U., Silalertruksa, T., & Gheewala, S. H. (2023). Optimization of food-fuel-fibre in biorefinery based on environmental and economic assessment: The case of sugarcane utilization in Thailand. Sustainable Production and Consumption, 37 , 398–411. https://doi.org/10.1016/j.spc.2023.03.013

Pornprakun, W., Sungnul, S., Kiataramkul, C., & Moore, E. J. (2019). Determining optimal policies for sugarcane harvesting in Thailand using bi-objective and quasi-newton optimization methods. Advances in Difference Equations, 2019 , 1–15.

Qiu, M., Meng, Y., Chen, J., Chen, Y., Li, Z., & Li, J. (2023). Dual multi-objective optimisation of the cane milling process. Computers & Industrial Engineering, 179 , 109146. https://doi.org/10.1016/j.cie.2023.109146

Ramos, R. P., Isler, P. R., Florentino, H. O., Jones, D., & Nervis, J. J. (2016). An optimization model for the combined planning and harvesting of sugarcane with maturity considerations. African Journal of Agricultural Research, 11 (40), 3950–3958.

Santoro, E., Soler, E. M., & Cherri, A. C. (2017). Route optimization in mechanized sugarcane harvesting. Computers and Electronics in Agriculture, 141 , 140–146.

Santos, J. A. (2022). Análise do sistema de energia fotovoltaica do Instituto Federal do Ceará-Campus Maracanaú e sua colaboração na redução da emissão de CO2. Available on: http://conexoes.ifce.edu.br/index.php/conexoes/article/view/2168 . Retrieved February 19, 2023.

Sethanan, K., & Neungmatcha, W. (2016). Multi-objective particle swarm optimization for mechanical harvester route planning of sugarcane field operations. European Journal of Operational Research, 252 (3), 969–984.

Shirvani, N., Ruiz, R., & Shadrokh, S. (2014). Cyclic scheduling of perishable products in parallel machine with release dates, due dates and deadlines. International Journal of Production Economics, 156 , 1–12.

Silva, A. F., Marins, F. A. S., & Dias, E. X. (2015). Addressing uncertainty in sugarcane harvest planning through a revised multi-choice goal programming model. Applied Mathematical Modelling, 39 (18), 5540–5558.

Toso, E. A., Morabito, R., & Clark, A. R. (2009). Lot sizing and sequencing optimisation at an animal-feed plant. Computers & Industrial Engineering, 57 (3), 813–821.

USDA: United States Department of Agriculture. (2023). Available on: https://apps.fas.usda.gov/psdonline/app/index.html#/app/advQuery . Retrieved January 31, 2023.

Usman, M., & Balsalobre-Lorente, D. (2022). Environmental concern in the era of industrialization: Can financial development, renewable energy and natural resources alleviate some load? Energy Policy, 162 , 112780.

Varshney, D., Mandade, P., & Shastri, Y. (2019). Multi-objective optimization of sugarcane bagasse utilization in an Indian sugar mill. Sustainable Production and Consumption, 18 , 96–114.

Walfrido, A. P., Luengo, C. A., Koehlinger, J., Garzone, P., & Cornacchia, G. (2008). Sugarcane energy use: The Cuban case. Energy policy, 36 (6), 2163–2181.

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The authors thank to Brazilian foundations: CNPq n \(^{\textrm{o}}\) 306518/2022-8, CNPq n \(^{\textrm{o}}\) 304218/2022-7, FAPESP 2021/03039-1,FAPESP 2022/12652-1, PROPE/PROPG/UNESP/ FUNDUNESP grant 12/2022, for the financial support and language services provided.

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Gilmar Tolentino, Antônio Roberto Balbo, Sônia Cristina Poltroniere, Angelo Aliano Filho and Helenice de Oliveira Florentino have contributed equally to this work.

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Department of Mathematics, State University of Sao Paulo, Bauru, São Paulo, 17033-360, Brazil

Gilmar Tolentino, Antônio Roberto Balbo & Sônia Cristina Poltroniere

Department of Mathematics, Universidade Tecnológica Federal do Paraná, Apucarana, Paraná, 86812-460, Brazil

Angelo Aliano Filho

Department of Bioestatistics, State University of Sao Paulo, Botucatu, São Paulo, 18618-690, Brazil

Helenice de Oliveira Florentino

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Tolentino, G., Balbo, A.R., Poltroniere, S.C. et al. A MIP-heuristic approach for solving a bi-objective optimization model for integrated production planning of sugarcane and energy-cane. Ann Oper Res (2024). https://doi.org/10.1007/s10479-024-06229-5

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Numerical algorithms for identification of convection coefficient and source in a magnetohydrodynamics flow.

1. Introduction

2. formulation of the direct and inverse problems, 2.1. the direct problem, 2.2. the inverse problems, 3. reduction of the ip1, ip2, and ip3 to nonclassical problems, 4. well-posedness of the inverse problems, 4.1. the ip2, 4.2. the ip1 and ip3, 5. finite difference approximation of the inverse problems, 6. algorithms for solution of finite difference algebraic systems, 6.1. the ip2, 6.2. the ip3 and ip1, 7. computational results, 7.1. inverse problem without noise, 7.2. inverse problem with a noise, 8. conclusions, author contributions, data availability statement, acknowledgments, conflicts of interest.

• Alifanov, O.M.; Artyukhin, E.A.; Rumyantsev, S.V. Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems ; Begell House: New York, NY, USA; Wallingford, UK, 2015. [ Google Scholar ]
• Canon, J.R.; Van Der Hoek, J. Diffusion subject to the specification of mass. J. Math. Appl. 1986 , 115 , 517–529. [ Google Scholar ] [ CrossRef ]
• Kabanikhin, S.I. Inverse and Ill-Posed Problems Theory and Applications ; Walter de Gruyter: Berlin, Germany, 2011. [ Google Scholar ]
• Krukovskiy, A.J.; Poveschenko, J.A.; Podruga, V.O. Convergence of some iterative algorithms for numerical solution of two-dimentional non-stationary problems of magnetic hydrodynamics. Math. Model. 2023 , 35 , 57–74. (In Russian) [ Google Scholar ]
• Ladyzhenskaia, V.A.; Solonnikov, V.A.; Ural’tceva, N.N. Linear and Quasi-linear Equations of Parabolic Type ; American Mathematical Society: Providence, RI, USA, 1986. [ Google Scholar ]
• Samarskii, A.A.; Vabishchevich, P.N. Numerical Methods for Solving Inverse Problems of Mathematical Physics ; Walter de Gruyter: Berlin, Germany, 2007. [ Google Scholar ]
• Vabishchevich, P.N.; Vasil’ev, V.I. Computational determination of the lowest order coefficient in a parabolic equation. Dokl. Math. 2014 , 89 , 179–181. [ Google Scholar ] [ CrossRef ]
• Vabishchevich, P.N.; Vasil’ev, V.I. Computational algorithms for solving the coefficient inverse problem for parabolic equations. Inv. Probl. Sci. Eng. 2016 , 24 , 42–59. [ Google Scholar ] [ CrossRef ]
• Ashyraliev, A.; Sazaklioglu, A.V. Investgation of a time-dependent source identification with integral overdetermination. Numer. Funct. Anal. Optim. 2017 , 38 , 1276–1294. [ Google Scholar ] [ CrossRef ]
• Borzi, A. Modelling with Ordinary Differential Equations. A Comprehensive Approach ; Chapman and Hall, CRC Press: Boca Raton, FL, USA, 2020. [ Google Scholar ]
• Georgiev, S.G.; Vulkov, L.V. Computational recovery of time-dependent volatility from integral observations in option pricing. J. Comput. Sci. 2019 , 39 , 101054. [ Google Scholar ] [ CrossRef ]
• Glotov, D.; Hames, W.E.; Meir, A.J.; Ngoma, S. An integral constrained parabolic problem with applications in thermochronology. Comput. Math. Appl. 2016 , 71 , 2301–2312. [ Google Scholar ] [ CrossRef ]
• Glotov, D.; Hames, W.E.; Meir, A.J.; Ngoma, S. An inverse diffusion coefficient problem for a parabolic equation with integral constraint. Int. J. Numer. Anal. Model. 2018 , 15 , 552–563. [ Google Scholar ]
• Evans, L.C. Partial Differential Equations , 2nd ed.; Graduate Studies in Mathematics 19; American Mathematical Society: Providence, RI, USA, 2010. [ Google Scholar ]
• He, Y. Unconditional convergence of the Euler semi-implicit scheme for the three dimensional incompressible MHD equations. IMA J. Numer. Anal. 2015 , 33 , 767–801. [ Google Scholar ] [ CrossRef ]
• Isakov, V. Inverse Problems for Partial Differential Equations , 2nd ed.; Springer: New York, NY, USA, 2006. [ Google Scholar ]
• Landau, L.D.; Bell, J.; Kearsley, M.; Pitaevski, L.; Lifshitz, E.; Sykes, J. Electrodynamics of Continuous Media ; Elsevier: Amsterdam, The Netherlands, 2013; eBook ISBN 9781483293752. [ Google Scholar ]
• Nguyen, T.N.O. Source identification for parabolic equations from integral observations by the finite difference splitting method. Acta Math. Vietnam. 2024 , 49 , 283–308. [ Google Scholar ]
• Samarskii, A.A.; Nikolaev, E.S. Numerical Methods for Grid Equations: Volume I Direct Methods ; Birkhäuser Verlag: Berlin, Germany, 1989. [ Google Scholar ]
• Slodicka, M.; Seliga, L. Identification of memory kernels in hyperbolic problems. J. Comp. Appl. Math. 2017 , 311 , 618–629. [ Google Scholar ] [ CrossRef ]
• Tsyba, V.; Chebatorev, A.Y. Optimal control asymptotics of a magnetohydrodynamic flow. Comp. Math. Math. Phys. 2009 , 49 , 466–473. [ Google Scholar ] [ CrossRef ]
• Marchuk, G.I. Adjoint Equations and Analysis of Complex Systems ; Kluwer Acad. Publishers: Dordrecht, The Netherlands, 1995. [ Google Scholar ]
• Koleva, M.N.; Vulkov, L.G. A Galerkin finite element method for reconstruction of time-dependent convection coefficients and source in a 1-D model of magnetohydrodynamics. Appl. Sci. 2024 , 14 , 5949. [ Google Scholar ] [ CrossRef ]
• Ren, Z.; Guo, S.; Li, Z.; Wu, Z. Adjoint-based parameter and state estimation in 1-D magnetohydrodynamics (MHD) flow system. J. Optim. Manag. Optim. 2018 , 14 , 1579–1594. [ Google Scholar ] [ CrossRef ]
• Yu, P.X.; Tian, Z.F. Comparison of the simplified and full MHD models for laminar incompressible flow past a circular cylinder. Appl. Math. Model. 2017 , 41 , 143–163. [ Google Scholar ] [ CrossRef ]
• Khankishiyev, Z.F. Solution of one problem for linear loaded parabolic type differential equation with integral conditions. Adv. Math. Mod. Appl. 2022 , 7 , 178–190. [ Google Scholar ]
• Vabishchevich, P.N.; Klibanov, M.V. Numerical identification of the leading coefficient of a parabolic equation. Differ. Equ. 2016 , 52 , 855–862. [ Google Scholar ] [ CrossRef ]
• Kandilarov, J.; Vulkov, L. Simultaneous numerical reconstruction of time-dependent convection coefficient and source in magnetohydrodynamics flow system. In Proceedings of the 16th Annual Meeting of the Bulgarian Section of SIAM, (BGSIAM’21) (to Appear), Sofia, Bulgaria, 21–23 December 2021. [ Google Scholar ]

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Kandilarov, J.D.; Vulkov, L.G. Numerical Algorithms for Identification of Convection Coefficient and Source in a Magnetohydrodynamics Flow. Algorithms 2024 , 17 , 387. https://doi.org/10.3390/a17090387

Kandilarov JD, Vulkov LG. Numerical Algorithms for Identification of Convection Coefficient and Source in a Magnetohydrodynamics Flow. Algorithms . 2024; 17(9):387. https://doi.org/10.3390/a17090387

Kandilarov, Juri D., and Lubin G. Vulkov. 2024. "Numerical Algorithms for Identification of Convection Coefficient and Source in a Magnetohydrodynamics Flow" Algorithms 17, no. 9: 387. https://doi.org/10.3390/a17090387

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