1. There are many ways to define decision variables in mathematical model. For a model formulated with on set of decision variables to be equivalent to another model of the same problem formulated using a different set of decision variables, some or all of the coefficients of the model will have to be different. Consider the problem faced by Shari Winslow.
Shari has $4000 which she wants to put into Keogh account before april 15. Doing so will save Shari over $1600 in federal and state tases.
Shari is not an experience inverstor, but she knows that she wants to invest in stocks rather than bonds or mutual funds. She has heard that financial expert, Craig Janes, has predicted large increases in the following stocks during the coming year:
· Shares of Topeka Electronics should rise to $50.
· Crosswind should experience a 20% rise in the value of its stock.
· Genserve should match the peak price of $75 it achieved during the past year.
The current share prices for Topeka Electronics, Crosswind, and Genserve are $42, $30, and $64, respectively. Shari has decided to invest in one or more of these companies for a year and then reevaluate her Keogh investments. While her goal is to maximize the projected return over the coming year, she has decided to be somewhat concervative and invest at least $800 in Crosswind and at least $1000 in a one-year savings certificate of deposit paying 5%. She wants to invest no more than $2000 in either Topeka Electronics or Genserve.
a. Formulate this problem as a mathematical model using decision variables X1, X2, and X3 to represent the dollar investment in Topeka Electronics, Crosswind, and Genserve, respectively, and X4 represents the amount invested in the savings certificate.
b. Formulate this problem as a mathematical model using decision variables Y1, Y2, and Y3, representing the number of shares stocks purchased in Topeka Electronics, Crosswind, and Genserve, respectively, with X4 still representing the amount invested in the savings certificate.
c. Suppose both the models in (a) and (b) were solved. How should the results compare?
2. The law of supply and demand states that if supply is limited and demand for a product is high, one should be able to charge a higher price for the product. Such was the case when the Mazda Miata sports car first was introduced. The base price (the sticker price) was about $13000. Mazda shipped the first few of the United States to build up interest in the car while larger quantities were being produced.
Because Mazda initially shipped so few Miatas to the dealers and customer demand was high, many dealers added a surcharge of up to $7000 to the sticker price. Many considered this action “gouging” while others simply called it “sound business practice.”
Suppose a management scientist has been able to forecast that if Springfield Motors charged $X (where X is in thousands of dollars) it could sell $4 – 2X Miatas in a month. Since each Miata costs Springfield Motors $11 (thousand), it will certainly not sell Miatas for less than $11.000. Mazda has informed Springfield Motors that if it sells Miatas for more than $21 (thousand), it will not ship any Miatas.
Write a model for Springfield Motors mothly revenue if it sells Miatas for $X (thousand). Remember : revenue equals selling price times the number of Miatas sold in a month. Note that this is a nonlinear function (i.e. it has a term with X raised to a power other than 1).
Write an optimization model for Springfield Motors monthly profit from selling Miatas subject to the limitations on price. Recall that profit = revenue – cost.
Using differential calculus, find the price X that maximizes the objective function in (b). Show that this price falls within the limits of the constraints. How many Miatas should Springfield Motors expect to sell each month at this price? What would be the monthly profit of this policy to Springfield Motors?
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