Math Modeling

In the fall 2012, I used the textbook "Mathematical Modeling" (3rd Edition) by Meerschaert (http://www.stt.msu.edu/~mcubed/modeling.html). Notice that the 4th edition has just come out. Overall this is a good book to introduce math modeling to undergraduates from different majors because the material is a nice balance of math theory, computer programming and applications to a variety of fields (such as biology, economics and physics). Following are some basic problems from this book that I required my students to master. Please refer to the book for details on the related theory and similar exercises.

1) Initial Value Problem

TYPE 1:
 
(Example 4.2) The blue whale and fin whale are two similar species that inhabit the same areas. Hence, they are thought to compete. The intrinsic growth rate of each species is estimated at 5% per year for the blue whale and 8% per year for the fin whale. The environmental carrying capacity (the maximum number of whales that the environment can support) is estimated at 150,000 blues and 400,000 fins. The extent to which the whales compete is unknown. In the last 100 years intense harvesting has reduced the whale population to around 5,000 blues and 70,000 fins.
 
*** Question 1: Build a dynamical system model to describe the changes of the whale populations. Use the number of blue and fin whales as state variables x1 and x2 (respectively) and make the simplest possible assumptions about growth and competition.
 

*** Question 2: To further simplify the problem, consider that the effect of competition is relatively small, such as 1e-7. Find the equilibrium points and plot the vector field for this problem. 

*** Question 3: Use the eigenvalue method to test the stability of each equilibrium. Can the two populations of whales grow to stable equilibrium starting from their current levels?

*** Question 4: If both populations of whales will eventually grow back to their natural levels in the absence of any further harvesting, how long will this take?

*** Question 5: What happens to the model if we use too large of a time step in the forward Euler method as we simulate the whale populations with respect to time?

Maple code (IVP.html)

Note: I will include some sensitivity analysis for this model in the future.

TYPE 2:

(Example 5.3) Consider the electrical circuit diagrammed in the following figure:

The circuit consists of a capacitor, a resistor, and an inductor in a simple closed loop. The effect of each component of the circuit is measured in terms of the relationship between current and voltage on that branch of the loop. An idealized

physical model gives the relations where vC represents the voltage across the capacitor, iR represents the current through the resistor, and so on. The function f(x) is called the v-i characteristic of the resistor. Usually f(x) has the same sign as x. This is called a passive resistor. Some control circuits use an active resistor, where f(x) and x have opposite sign for small x. In the classical linear model of the RLC circuit, we assume that f(x) = Rx where R > 0 is the resistance. Kirchoff’s current law states that the sum of the currents flowing into a node equals the sum of the currents flowing out. Kirchoff’s voltage law states that the sum of the voltage drops along a closed loop must add up to zero. Determine the behaviors of this circuit over time under the effect of varying the capacitance C over the entire range 0 < C < infinity, and f(x) = x3 + 4x by:

*** sketching the phase portrait for the linear system to illustrate the equilibrium and solution curves.

Maple code (Q11_C5.html)

*** solving the linear system by the method of eigenvalues and eigenvectors. 

Maple code (Q11bis_C5.html)

2) Unconstrained Optimization 
 
(Example 2.1) A manufacturer of color TV sets is planning the introduction of two new products, a 19–inch LCD flat panel set with a manufacturer’s suggested retail price (MSRP) of $339 and a 21–inch LCD flat panel set with an MSRP of $399. The cost to the company is $195 per 19–inch set and $225 per 21–inch set, plus an additional $400,000 in fixed costs. In the competitive market in which these sets will be sold, the number of sales per year will affect the average selling price. It is estimated that for each type of set, the average selling price drops by one cent for each additional unit sold. Furthermore, sales of the 19– inch set will affect sales of the 21–inch set, and vice–versa. It is estimated that the average selling price for the 19–inch set will be reduced by an additional 0.3 cents for each 21–inch set sold, and the price for the 21–inch set will decrease by 0.4 cents for each 19–inch set sold. How many units of each type of set should be manufactured?
*** Part 1: Use 3D graph and contour plot of profit versus production levels of 19–inch sets and 21–inch sets to answer the question.
 
 
*** Part 2: Use a computer algebra system to answer to the question.
 
*** Part 3: Since this is an optimization problem, we'll need to find critical numbers, i.e., set the gradient of the function equal to zero. Now we're dealing with the root-finding problem: we can use Newton's method to identify the locations of these critical numbers. Notice that the method of random search can be utilized to find an initial approximate root for Newton's method. 
 

Maple code (Q7_C3a.html)

Note: I will include some sensitivity analysis for this problem in the future.

3) Constrained Optimization: Lagrange multipliers 

(Example 2.2) Reconsider the color TV problem above. There we assumed that the company has the potential to produce any number of TV sets per year. Now we will introduce constraints based on the available production capacity. Consideration of these two new products came about because the company plans to discontinue manufacture of some older models, thus providing excess capacity at its assembly plant. This excess capacity could be used to increase production of other existing product lines, but the company feels that the new products will be more profitable. It is estimated that the available production capacity will be sufficient to produce 10,000 sets per year (≈ 200 per week). The company has an ample supply of 19–inch and 21–inch LCD panels and other standard components; however, the circuit boards necessary for constructing the sets are currently in short supply. Also, the 19–inch TV requires a different board than the 21–inch model because of the internal configuration, which cannot be changed without a major redesign, which the company is not prepared to undertake at this time. The supplier is able to supply 8,000 boards per year for the 21–inch model and 5,000 boards per year for the 19–inch model. Taking this information into account, how should the company set production levels?

*** Part 1: Use the method of Lagrange multipliers to answer the question. Since the Lagrange multipliers represent shadow prices, interpret this number in the context of the TV problem.
 
 
***Part 2: What can be a disadvantage of using the method of Lagrange multipliers?

Note: I will include some sensitivity analysis for this problem in the future.

4) Linear Programming

(Example 3.4) A family farm has 625 acres available for planting. The crops the family is considering are corn, wheat, and oats. It is anticipated that 1,000 acre–ft of water will be available for irrigation, and the farmers will be able to devote 300 hours of labor per week. Additional data are presented below.

Use a computer algebra system to answer to the following questions:

*** Question 1: Find the amount of each crop that should be planted for maximum profit.  

*** Question 2: What will be the effect of one additional acre–ft of irrigation water on the optimal solution?

*** Question 3: What will be the effect of a slightly higher yield for corn ($450 instead of $400) on the optimal solution?

*** Question 4: What will be the effect of a slightly higher yield for oats ($260 instead of $250) on the optimal solution?