-These problems focus on finding solutions to numerical problems. With that in mind, most problem sets will include a number of problems. For each problem, you will need to provide more than a simple numerical response. Your solutions should thoroughly address the issue and present your findings in a meaningful format.
1- Discuss the different roles played by the qualitative and quantitative approaches to managerial decision making. Why is it important for a manager or decision maker to have a good understanding of both of these approaches to decision making?
2- What are the advantages of analyzing and experimenting with a model as opposed to a real object or situation?
3- Suppose you are going on a weekend trip to a city that is d miles away. Develop a model that determines your round-trip gasoline costs. What assumptions or approximations are necessary to treat this model as a deterministic model? Are these assumptions or approximations acceptable to you?
4- A retail store in Des Moines, Iowa, receives shipments of a particular product from Kansas City and Minneapolis. Let x 5 units of product received from Kansas City y 5 units of product received from Minneapolis
a. Write an expression for the total units of product received by the retail store in Des Moines. b. Shipments from Kansas City cost $0.20 per unit, and shipments from Minneapolis cost $0.25 per unit. Develop an objective function representing the total cost of shipments to Des Moines.
c. Assuming the monthly demand at the retail store is 5000 units, develop a constraint that requires 5000 units to be shipped to Des Moines.
d. No more than 4000 units can be shipped from Kansas City and no more than 3000 units can be shipped from Minneapolis in a month. Develop constraints to model this situation.
e. Of course, negative amounts cannot be shipped. Combine the objective function and constraints developed to state a mathematical model for satisfying the demand at the Des Moines retail store at minimum cost.
5- For most products, higher prices result in a decreased demand, whereas lower prices result in an increased demand (economists refer to such products as normal goods). Let
d = 5 annual demand for a product in units.
p= 5 price per unit .
Assume that a firm accepts the following price–demand relationship as being a real- istic representation of its market:
d =5 800 – 10p
where p must be between $20 and $70.
a. How many units can the firm sell at the $20 per-unit price? At the $70 per-unit price?
b. What happens to annual units demanded for the product if the firm increases the per- unit price from $26 to $27? From $42 to $43? From $68 to $69? What is the suggested relationship between per-unit price and annual demand for the product in units?
c. Show the mathematical model for the total revenue (TR), which is the annual demand multiplied by the unit price.
d. Based on other considerations, the firm’s management will only consider price alternatives of $30, $40, and $50. Use your model from part (b) to determine the price alternative that will maximize the total revenue.
e. What are the expected annual demand and the total revenue according to your recommended price?
6- Micromedia offers computer training seminars on a variety of topics. In the seminars each student works at a personal computer, practicing the particular activity that the instructor is presenting. Micromedia is currently planning a two-day seminar on the use of Microsoft Excel in statistical analysis. The projected fee for the seminar is $600 per student. The cost for the conference room, instructor compensation, lab assistants, and promotion is $9600. Micromedia rents computers for its seminars at a cost of $60 per computer per day.
a. Develop a model for the total cost to put on the seminar. Let x represent the number of students who enroll in the seminar.
b. Develop a model for the total profit if x students enroll in the seminar.
c. Micromedia has forecasted an enrollment of 30 students for the seminar. How much profit will be earned if its forecast is accurate?
d. Compute the break-even point.
7- Financial Analysts, Inc., is an investment firm that manages stock portfolios for a number of clients. A new client has requested that the firm handle an $800,000 portfolio. As an initial investment strategy, the client would like to restrict the portfolio to a mix of the following two stocks:
STOCK Price/ Estimated Annual
Oil Alaska $50 $6
Southwest Petroleum $30 $4
x= number of shares of Oil Alaska
y= number of shares of Southwest Petroleum
a- Develop the objective function, assuming that the client desires to maximize the total annual return.
b- Show the mathematical expression for each of the following three constraints: (1) Total investment funds available are $800,000. (2) Maximum Oil Alaska investment is $500,000. (3) Maximum Southwest Petroleum investment is $450,000.
Note: Adding the x $ 0 and y $ 0 constraints provides a linear programming model for the investment problem.
8- Models of inventory systems frequently consider the relationships among a beginning inventory, a production quantity, a demand or sales, and an ending inventory. For a given production period j, let
S j–1 = beginning inventory for period j (ending inventory from period j – 1, the previous period)
x j = production quantity in period j
d j = demand in period j
s j = ending inventory for period j
a. Write the mathematical relationship or model that shows ending inventory as a function of beginning inventory, production, and demand.
b. What constraint should be added if production capacity for period j is given by Cj?
c. What constraint should be added if inventory requirements for period j mandate an ending inventory of at least I j?