Question: A company uses raw material to produce

> Modify the function in Example 8.1 so that it becomes f(x) = x sin(x) for 0 ≤ x ≤ 30. (Here, sin(x) is the sine function from trigonometry. You can evaluate it with Excel’s SIN function.) Plot a lot of points from 0 to 30 to see what the graph of this fu

> The 30 teams in the NBA are each assigned to one of six divisions, where each division has five teams. Suppose the goal is to assign the teams to divisions so that the average distance among teams in the divisions is minimized. In other words, the goal i

> You are operating a Web site to match up sellers and buyers of a product. 35 sellers and 35 buyers have input their reservation prices, as listed in the file. For example, buyer 1 is willing to pay up to $8 for an item, and seller 1 is willing to accept

> Repeat Problem 15, but now assume there is a setup time for changing from any job to another job, and this setup time can depend on the jobs. For example, the setup time when changing from job 2 to job 4 can be different from the setup time when changing

> The traveling salesperson problem is notoriously difficult when the number of cities is even of moderate size. The file contains two sheets, one with a distance matrix for a 30-city problem and the other with a distance matrix for a 45-city problem. See

> An important problem in operations management is the job sequencing problem. Actually, there are many versions of this problem, but they all basically attempt to find the proper sequencing of jobs on a machine. Here is one version of the problem. There a

> Continuing the previous problem, determine the portfolio that minimizes the chance that you will lose money during any month, subject to a lower bound constraint on your expected monthly return. (The lower bound will depend on your data. It must not be a

> Visit biz.yahoo.com/r/. Under Research Tools, click on Historical Quotes, and then download the monthly returns on at least four stocks for the preceding 60 months. Use this data to determine the portfolio that maximizes the chance of beating the S&P 500

> Sales of a product over time often follow an S-shaped curve. Two functions that yield S-shaped curves are the Pearl (or logistic) curve Y = L / + ae-bt and the Gompertz curve Y = Le-be-kt Here, Y is annual sales, t is time (in years), L is the upper lim

> In a typical production scheduling model like Pigskin’s, if there are no production capacity constraints—the company can produce as much as it needs in any time period—but there are storage capacity constraints and demand must be met on time, is it possi

> The adoption level of a new product can often be modeled as an S-shaped curve called the Pearl (or logistic) curve. The equation of this curve is Y = L / + ae-bt where Y is the adoption level, L is an (unknown) upper limit on adoptions, t is time (in ye

> The file contains per capita data on annual advertising and annual unit sales in different regions of the country. Determine an S-shaped curve as described by Equation that can be used to determine how advertising influences sales.

> Modify the function in Example 8.1 so that it becomes f(x) = (x - 1)(x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7) for 1 ≤ x ≤ 7. Plot a lot of points from 1 to 7 to see what the graph of this function looks like. Then use GRG Nonlinear Solver to find its ma

> In the complementary-product pricing model, the elasticity of demand for suits is currently 22.5. Use SolverTable to see how the optimal price of suits and the optimal profit vary as the elasticity varies from 22.7 to 21.8 in increments of 0.1. Are the r

> Your family owns a large farm that can grow wheat, corn, cotton, alfalfa, barley, pears, and apples. Each product requires a certain amount of labor each month and a certain number of hours of machine time. You have just studied portfolio optimization an

> UE is going to invest $400 million to acquire companies in the auto and/or electronics industry. How would you apply portfolio optimization to determine which companies should be purchased?

> Consider the sports ratings model. If you were going to use the approach used there to forecast future sports contests, what problems might you encounter early in the season? How might you resolve these problems?

> Consider the sports ratings model. If you were going to give more recent games more weight, how might you determine whether the weight given to a game from k weeks ago should be, say, (0.95)k or (0.9)k?

> For the aggregate planning example, is it likely that the cost per worker of changing the size of the workforce during a month would be constant (as we assumed)? How could an NLP model account for a situation in which the cost per worker of changing the

> For the oil blending example, discuss where you think the assumptions of a linear model are most likely to break down. How might an NLP model look in this situation?

> In a typical product mix model, where a company must decide how much of each product to produce to maximize profit, there are sometimes customer demands for the products. We used upper-bound constraints for these: Don’t produce more than you can sell. Wo

> In some ordering problems, like the one for Sam’s Bookstore, whenever demand exceeds existing inventory, the excess demand is not lost but is filled by expedited orders—at a premium cost to the company. Change Sam’s model to reflect this behavior. Assume

> For the product mix examples, discuss where you think the assumptions of a linear model are most likely to break down. How might an NLP model look in this situation?

> In the exchange rate model, we found that the optimal unit revenue, when converted to dollars, is $85.71. Now change the problem so that the company is selling in Japan, not the United Kingdom. Assume that the exchange rate is 0.00965 ($/¥) and that the

> Your company is about to market a new golf club. You have convened a focus group of 100 golfers and asked them to compare your club to the clubs produced by your competitors. You have found, for example, that 30 customers in the focus group would purchas

> A triangle has a 5-inch side and a 12-inch side. To maximize the area of the triangle what should the third side be? Can you generalize this result?

> You can swim two miles per hour and run six miles per hour. You are walking north along South Beach and see someone drowning half a mile out in the ocean and one mile north of you. What combination of running and swimming is the quickest way to reach the

> City B is 10 miles downstream from city A. City A is 5 miles south of the river, and city B is 20 miles north of the river. The river is two miles wide. Where should a bridge be built across the river to make the travel distance between cities A and B as

> A cylindrical soda can has a volume of 20 cubic inches. What height and diameter minimize the surface area of the can? Can you generalize this result?

> You are given that the two non-hypotenuse sides of a right triangle add up to 10 inches. What is the maximum area of the triangle? Can you generalize this result?

> Find the minimum perimeter rectangle having area 64 square feet. Can you generalize this result?

> Kellpost Cereal Company sells four products: (1) Special L (a low-calorie, high-nutrition cereal); (2) Corn Bran (another low-calorie, high-nutrition cereal); (3) Admiral Smacks (a sugary cereal pitched at the children’s market); and (4) Honey Pops (anot

> In a typical product mix model, where a company must decide how much of each product to produce to maximize profit, discuss possible situations where there might not be any feasible solutions. Could these be realistic? If you had such a situation in your

> Based on Grossman and Hart (1983). A salesperson for Fuller Brush has three options: (1) quit, (2) put forth a low level of effort, or (3) put forth a high level of effort. Suppose for simplicity that each salesperson will sell $0, $5000, or $50,000 wort

> Each morning during rush hour, 10,000 people want to travel from New Jersey to New York City. If a person takes the commuter train, the trip lasts 40 minutes. If x thousand people per morning drive to New York, it takes 20 1 5x minutes to make the trip.

> Change the exchange rate model slightly so that the company is now a UK manufacturing company producing for a U.S. market. Assume that the unit cost is now £75, the demand function has the same parameters as before (although the price for this demand fun

> Most economies have a goal of maximizing the average consumption per period. Assume that during each year, an economy saves the same (to be determined) percentage S of its production. During a year in which the beginning capital level is K, a quantity K1

> Four items are for sale in the Dollar Value store. The product and sum of their prices is $7.11. What is the price of each item?

> The cost of producing x units of a product during a month is x1/2 dollars. Show that the minimum cost method of producing 40 units during the next two months is to produce all 40 units during a single month.

> Suppose that you are hiring a weather forecaster to predict the probability that next summer will be rainy or sunny. The following suggests a method that can be used to ensure that the forecaster is accurate. Suppose that the actual probability of next s

> An oil company must determine how many barrels of oil to extract during each of the next two years. If the company extracts x1 million barrels during year 1, each barrel can be sold for 80 - x1 dollars. If the company extracts x2 million barrels during y

> You have $50,000 to invest in three stocks. Let Ri be the random variable representing the annual return on $1 invested in stock i. For example, if Ri = 0.12, then $1 invested in stock i at the beginning of a year is worth $1.12 at the end of the year. T

> Consider three investments. You are given the following means, standard deviations, and correlations for the annual return on these three investments. The means are 0.12, 0.15, and 0.20. The standard deviations are 0.20, 0.30, and 0.40. The correlation b

> Suppose you have a linear optimization model where you are trying to decide which products to produce to maximize profit. What does the additive assumption imply about the profit objective? What does the proportionality assumption imply about the profit

> Monroe County is trying to determine where to place the county fire station. The locations of the county’s four major towns are as follows: (10, 20), (60, 20), (40, 30), and (80, 60) (see Figure 7.50). Town 1 averages 40 fires per year;

> A company has five factories. The x- and y-coordinates of the location of each factory are given in the file. The company wants to locate a warehouse at a point that minimizes the sum of the squared distances of the plants from the warehouse. Where shoul

> Based on Kolesar and Blum (1973). Suppose that a company must service customers lying in an area of A square miles with n warehouses. Kolesar and Blum showed that when the warehouse(s) are located properly, the average distance between a warehouse and a

> In the exchange rate model, suppose the company continues to manufacture its product in the United States, but now it sells its product in the United States, the United Kingdom, and possibly other countries. The company can independently set its price in

> J&J has given you $12 million to spend on advertising Huggys diapers during the next 12 months. At the beginning of January, Huggys has a 30% market share. During any month, 10% of the people who purchase Huggys defect to brand X, and a fraction 0.2a1/2

> A beer company has divided Bloomington into two territories. If the company spends x1 dollars on promotion in territory 1, it can sell 60x1/21 cases of beer there; and if it spends x2 dollars on promotion in territory 2, it can sell 40x1/22 cases of beer

> Q&H Company advertises during soap operas and football games. Each soap opera ad costs $75,000, and each football game ad costs $130,000. If S soap opera ads are purchased, they will be seen by 4S1/2 million men and 22S1/2 million women. If F football ad

> A brewing company has $100,000 to spend on advertising in four markets. The sales revenue (in thousands of dollars) that can be created in each market by spending xi thousand dollars in market i is given in the file. a. To maximize its sales revenue, how

> A company manufactures two products. If it charges price pi for product i, it can sell qi units of product i, where q1 = 80 - 2p1 + p2 and q2 = 100 - 3p2 + p1. It costs $8 to produce a unit of product 1 and $15 to produce a unit of product 2. How many un

> Why is it generally necessary to add nonnegativity constraints to an optimization model? Wouldn’t Solver automatically choose nonnegative values for the decision variable cells?

> Two companies are producing widgets. It costs the first company q21 dollars to produce q1 widgets and the second company 0.75q22 dollars to produce q2 widgets. If a total of q widgets are produced, consumers will pay 300 - q dollars for each widget. If

> The cost per day of running a hospital is 300,000 + 0.75x2 dollars, where x is the number of patients served per day. What number of patients served per day minimizes the cost per patient per day of running the hospital if the hospital’s daily capacity i

> In the previous problem, what is the minimum-cost method of producing 100 machines? (Ignore the $10,000 budget constraint.)

> It costs a company $15 to purchase an hour of labor and $20 to purchase an hour of capital. If L hours of labor and K units of capital are available, then 0.075L2/3K1/3 machines can be produced. Suppose the company has $150,000 to purchase labor and capi

> Continuing Problem 3 in a slightly different direction, create a two-way SolverTable where the inputs are the elasticity and the production capacity, and the outputs are the optimal price and the optimal profit. (This actually creates two tables, one for

> If a monopolist produces q units, she can charge 400 - 4q dollars per unit. The variable cost is $60 per unit. a. How can the monopolist maximize her profit? b. If the monopolist must pay a sales tax of 5% of the selling price per unit, will she increase

> A company produces widgets at plants 1 and 2. It costs 135x1/2 dollars to produce x units at plant 1 and 260x1/3 dollars to produce x units at plant 2. Each plant can produce up to 1000 units. Each unit produced can be sold for $12. At most 1700 widgets

> Another way to derive a demand function is to break the market into segments and identify a low price, a medium price, and a high price. For each of these prices and market segments, we ask company experts to estimate product demand. Then we use Excel’s

> Suppose the current exchange rate is 115 yen per dollar. We currently have a demand for 50 units of our product when the unit price is 800 yen. The cost of producing and shipping the product to Japan is $6, and the current elasticity of demand is -2.5. F

> Suppose Ford currently sells 250,000 Ford Mustangs annually. The unit cost of a Mustang, including the delivery cost to a dealer, is $16,000. The current Mustang price is $20,000, and the current elasticity of demand for the Mustang is 21.5. a. Determine

> The file contains data on prices of products for several of a chain store’s locations, a discount schedule offered to customers depending on how much they spend, and commission rates of the salespeople at the various stores. Your job is to develop an inv

> Solve the previous problem, but analyze GE instead of Microsoft.

> Given the data in the file Stock Beta.xlsx, estimate the beta (and alpha) for Microsoft (MSFT). Do this for each criterion to obtain a table analogous to that in the top right. What do you conclude about Microsoft?

> In many cases, the portfolio return is at least approximately normally distributed. Then Excel’s NORMDIST function can be used to calculate the probability that the portfolio return is negative. The relevant formula is =NORMDIST(0,mean, stdev,True), wher

> Continuing the previous problem, find the portfolio that achieves an expected monthly return of at least 1% and minimizes portfolio variance. Then use SolverTable to sweep out the efficient frontier. Create a chart of this efficient frontier from your So

> This problem continues using the data from the previous problem. The file includes all of the previous data and contains investment weights in row 3 for creating a portfolio. These fractions are currently all equal to 1/27, but they can be changed to any

> Continuing the previous problem, create a two-way data table similar to the one-way data. This time, however, allow price to vary down a column and allow the capacity to vary across a row. Each cell of the data table should capture the corresponding prof

> The file contains historical monthly returns for 27 companies. For each company, calculate the estimated mean return and the estimated variance of return. Then calculate the estimated correlations between the companies’ returns. Note that “return” here m

> The stocks are all positively correlated. What happens when they are negatively correlated? Answer for each of the following scenarios. In each case, two of the three correlations are the negatives of their original values. Discuss the differences betwee

> In the model, stock 2 is not in the optimal portfolio. Use SolverTable to see whether it ever enters the optimal portfolio as its correlations with stocks 1 and 3 vary. Specifically, use a two-way SolverTable with two inputs, the correlations between sto

> Add a new stock, stock 4, to the model. Assume that the estimated mean and standard deviation of return for stock 4 are 0.125 and 0.175, respectively. Also, assume the correlations between stock 4 and the original three stocks are 0.3, 0.5, and 0.8. Run

> A project does not necessarily have a unique IRR. (Refer to the previous problem for more information on IRR.) Show that a project with the following cash flows has two IRRs: year 1, -$20; year 2, $82; year 3, -$60; year 4, $2.

> For each of the following, answer whether it makes sense to multiply the matrices of the given sizes. In each case where it makes sense, demonstrate an example in Excel, where you can make up the numbers. a. AB, where A is 3 × 4 and B is 4 × 1 b. AB, whe

> The method for rating teams is based on actual and predicted point spreads. This method can be biased if some teams run up the score in a few games. An alternative possibility is to base the ratings only on wins and losses. For each game, you observe whe

> By the time you are reading this, the 2016–2017 NBA season will have finished, and the results should be available at www.basketball-reference.com/leagues/NBA_2017_games.html. Do whatever it takes to get the data into Excel in the format of this example.

> By the time you are reading this, the 2016 NFL season will have finished, and the results should be available at www.pro-football-reference.com/years/ 2016/games.htm. Do whatever it takes to get the data into Excel in the format of this example. Then, us

> The file P07_31.xlsx contains scores on all of the regular-season games in the NBA for the 2015–2016 basketball season. Use the same procedure as in Example 7.8 to rate the teams. Then sort the teams based on the ratings. Do these ratings appear to be ap

> Carry out the suggestion in Modeling Issue 2 for the 2015 NFL season. That is, use a weighted sum of squared prediction errors, where the weight on any game played k weeks ago is 0.95k. You can assume that the ratings are being made right after the final

> In the pricing model with the constant elasticity demand function, the assumption is that all units demanded are sold. Suppose the company has the capacity to produce only 200 units. If demand is less than capacity, all of demand is sold. If demand is gr

> Carry out the suggestion in Modeling Issue 3. That is, find the ratings of the 2015 NFL teams using the sum of absolute prediction errors as the criterion to minimize. Discuss any differences in ratings from this method and the method.

> The file P07_28.xlsx lists the scores of all NFL games played during the 2014 season. Use this data set to rank the NFL teams from best to worst.

> Modify the warehouse location model as suggested in Modeling Issue 2. Specifically, assume that the same four customers have the same annual shipments, but now, there are only two possible warehouse locations, each with distances to the various customers

> The IRR is the discount rate r that makes a project have an NPV of $0. You can find IRR in Excel with the built-in IRR function, using the syntax =IRR(range of cash flows). However, it can be tricky. In fact, if the IRR is not near 10%, this function mig

> Use SolverTable in the warehouse location model to see the effect on the optimal solution of moving one customer farther and farther away from the others. Specifically, let customer 1’s coordinates be of the form (5c, 10c), where the factor c is allowed

> Modify the warehouse location model so that customers always travel in horizontal or vertical directions. For example, this means that if a customer’s coordinates are (5, 10) and a warehouse is located at (7, 7), then the traveling distance is |5 – 7| +

> Modify the warehouse location model so that there is an extra customer. This customer has 250 shipments per year. Try placing this new customer at various locations. For example, try placing the customer way up to the right, or way down to the left, or n

> We implied that each of the five observations was from one period of time, such as a particular week. Suppose instead that each is an average over several weeks. For example, the 4.7 million exposures corresponding to one ad might really be an average ov

> The advertising response function is only one of several nonlinear functions that could be used to get the same “increasing at a decreasing rate” behavior. Another possibility is the function f (n) = anb, where a and b are again constants to be determine

> In the solution to the advertising selection model, we indicated that the women 36 to 55 group is a bottleneck in the sense that the company needs to spend a lot more than it would otherwise have spent to meet the constraint for this group. Use SolverTab

> The preceding problem indicates how fewer alternatives can cause total cost to increase. This problem indicates the opposite. Starting with the solution to the advertising selection problem in Example 7.6, add a new show, “The View,” which appeals primar

> One demand function is linear and the other is called a constant elasticity demand function. Using data tables, show that the price elasticity in the linear demand function is not constant in price, and show that the price elasticity is constant in the c

> Starting with the solution to the advertising selection problem, suppose the company, for whatever reason, cannot place ads on “SportsCenter.” Make the appropriate changes in the model and rerun Solver. Comment on the changes to the decision variable cel

> In judging the fit of the estimated response function, you could use MAE (mean absolute error) instead of RMSE. MAE is the average of the absolute prediction errors. a. When you run Solver with MAE as your objective, do you get approximately the same est

> The file contains a template for a car loan. Specifically, once values are entered in the blue cells, you need to enter formulas in the gray cells to calculate the amount financed, the monthly payment (assuming that monthly payments stay the same through

> The best-fitting advertising response function fits the observed data. This is because we chose the observed data to fall close to a curve of the form. See what happens when one of the observed points is an outlier—that is, it doesn’t fit the pattern of

> In estimating the advertising response function, we indicated that the sum of squared prediction errors or RMSE could be used as the objective, and we used RMSE. Try using the sum of squared prediction errors instead. Does Solver find the same solution a