DS631 Final Exam Fall, 2016 Dr. F. V. Lu Please use the provided shell file: DS631 Final Exam-Shell.xls to complete the following five problems. Problem 1 - Integer (Binary) Programming A company is considering five investments. The initial cash outflow (cost) and net present value (yield) for each investment are given on Problem 1 sheet of the shell file. The cash available for investment is $13,000. Note that if the company wishes to take part in any of these investments, it must \"go all the way.\" In other words, partial investments are not permitted. Formulate and solve an appropriate integer programming model to help this company identify a feasible investment plan that maximizes total net present value. Problem 2 - Lockboxes A national retail chain receives credit card payments from four regions of the country (West, Midwest, East, and South). The average daily payments from each region are given on the Problem 2 sheet of the shell. This company's management must decide where customers should mail their payments. An annual interest rate of 18% can be earned on cash received. The company is considering setting up payment centers in four cities: San Francisco, St. Louis, Philadelphia, and Tampa. The average number of days that elapse between the time a payment check is mailed and the time the check is credited to one of the company's accounts is also given on Problem 2 sheet of the shell file. The annual cost of operating a payment center is $100,000. Moreover, each payment center that is opened could handle at most $270,000 per day in payments. All of the customers in each region must send all of their payments to a single payment center. This company wants to determine the payment center configuration that minimizes the sum of annual lost interest and center operating costs. Using the shell provided on Problem 2 sheet, formulate and solve an integer programming model to help this retailer solve its decision problem. Problem 3 - Constant Elasticity Demand A company currently sells 1000 units of its most popular product each week. The unit cost of producing and marketing this product is $25. The current selling price is $50, and the current elasticity of demand is -2.0. Assume that the demand function for this product is of the constant elasticity type. This company wants to find the profit-maximizing price, demand, and profit levels. Formulate and solve an appropriate nonlinear optimization model on Problem 3 sheet of the shell file to help this company identify the best price for the product. Problem 4 - Least Squares Regression A toy manufacturer has observed the weekly production and weekly variable production cost of a small yet very popular product at its Kansas City plant. These data are recorded on the Problem 4 sheet of the shell file. Management of this toy company believes that weekly production and weekly variable costs are related as follows: Weekly variable cost = + Weekly production + Error term, where and are constants. Formulate and solve an appropriate nonlinear optimization model on Problem 4 sheet of the shell file to estimate this equation using the given data with the least squares criterion. Problem 5 - minimum-variance portfolio A recent graduate from an MBA program is trying to decide how to allocate her available funds among three investment opportunities: stock 1, stock 2, and stock 3. From past data, the means and standard deviations of annual returns have been estimated as shown on the Problem 5 sheet of the shell file. The correlations between the annual returns of the stocks are also listed in the given file. This investor wants to find a minimum-variance portfolio that yields an expected annual return of at least 14%. Formulate and solve an appropriate nonlinear optimization model on Problem 5 sheet of the shell file to solve this problem. Problem 6.1A Input data on potential investments Investment Investment cost NPV 1 $4,500 $20,000 2 $3,500 $12,000 Amt invested Budget $13,000 Investment levels Budget constraint Total NPV 3 $5,000 $12,000 4 $5,500 $16,000 5 $4,000 $18,000 Problem 6.11A Financial data Annual interest rate Cost per lockbox 18% $100,000 Days in transit Lockbox location Check origin West Midwest East South San Francisco 3 4 5 4 St. Louis Philadelphia 3 4 3 3 4 3 4 4 Tampa 5 4 3 3 Payments per day $115,000 $125,000 $140,000 $130,000 Annual lost interest for origin/location pair Lockbox location Check origin West Midwest East South San Francisco St. Louis Philadelphia Tampa St. Louis Philadelphia Tampa St. Louis Philadelphia Tampa Assignments and lockbox decisions Locations of lockboxes San Francisco Assignments To lockbox location Sent from West Midwest East South San Francisco <= Constraint on amount handled per day San Francisco Total handled per day Maximum Monetary summary Lost interest Lockbox cost $270,000 St. Louis Philadelphia $270,000 $270,000 Tampa $270,000 Sent? Required Total cost Payments per day Problem 7.1A Unit cost Current price Current demand $25 $50 1000 Parameters of constant elasticity demand function Constant Elasticity -2.000 Pricing model Optimal price Optimal demand Profit Problem 7.9A Parameters Alpha Historical data Week 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Production 2000 4000 7000 1000 9000 5000 3000 2500 10000 7500 6000 4500 3000 8000 8500 Sum of squared errors Beta Variable cost Predicted cost $6,000 $11,500 $23,000 $4,000 $26,300 $16,150 $8,560 $7,300 $29,175 $24,060 $18,090 $13,110 $9,245 $25,685 $27,100 Error Squared error Problem 7.12A Stock input data Mean return StDev of return Stock 1 0.15 0.16 Stock 2 0.14 0.12 Stock 3 0.12 0.10 Correlations Stock 1 Stock 2 Stock 3 Stock 1 1 0.8 -0.7 Stock 2 0.8 1 0.5 Stock 3 -0.7 0.5 1 Stock 1 Stock 2 Stock 3 Covariances Stock 1 Stock 2 Stock 3 Investment decisions Fractions to invest Constraint on expected portfolio return Actual Required 0.14 Portfolio variance Total Required Stock 1 Stock 2 Stock 3