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practical management science
Practical Management Science, Revised 3rd Edition Wayne L Winston, S. Christian Albright - Solutions
If a monopolist produces q units, she can charge 100 4q dollars per unit. The fixed cost of production is $50 and the variable cost is $2 per unit.a. How can the monopolist maximize her profit?b. If the monopolist must pay a sales tax of $2 per unit, will she increase or decrease
Each plant can produce up to 70 units. Each unit produced can be sold for $10. At most 120 widgets can be sold. Determine how Widgetco can maximize its profit.
It costs 20x12 dollars to produce x units at plant 1 and 40x13 dollars to produce x units at plant
Widgetco produces widgets at plants 1 and
Suppose Ford currently sells 250,000 Ford Mustangs annually. The unit cost of a Taurus, 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 1.5.a. Determine a profit-maximizing price for a Mustang. Do
Repeat the previous problem, but analyze IBM instead of GM.
Given the data in the file Stock Beta.xlsx, estimate the beta (and alpha) for GM. Do this for each criterion and each period of time to obtain a table analogous to that in Figure 7.40. What do you conclude about GM?
In many cases, we can assume that the portfolio return is at least approximately normally distributed. Then we can use Excel’s NORMDIST function to calculate the probability that the portfolio return is negative. The relevant formula is =NORMDIST(0,mean,stdev,1), where mean and stdev are the
Continuing the previous problem, find the portfolio that achieves an expected monthly return of at least 0.01% and minimizes portfolio variance. Then use SolverTable to sweep out the efficient frontier, as in Example 7.9. Create a chart of this efficient frontier from your SolverTable results. What
For any such fractions, find the estimated mean, variance, and standard deviation of the resulting portfolio return.Skill-Extending Problems
This problem continues using the data from the previous problem. The file P07_40.xlsx includes all of the previous data and contains fractions in row 3 for creating a portfolio. These fractions are currently all equal to 1/28, but they can be changed to any values you like, as long as they continue
The file P07_39.xlsx contains historical monthly returns for 28 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 means monthly return. (Hint:
(Here, stock 3 tends to go in a different direction from stocks 1 and 2.)
(Here, stock 2 tends to go in a different direction from stocks 1 and 3.)c. Change the signs of the correlations between stocks 1 and 3 and between stocks 2 and
(Here, stock 1 tends to go in a different direction from stocks 2 and 3.)b. Change the signs of the correlations between stocks 1 and 2 and between stocks 2 and
The stocks in Example 7.9 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 between the optimal portfolios in
In the model in Example 7.9, 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 stock 2 and stocks 1 and 3, each
Add a new stock, stock 4, to the model in Example 7.9. 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 Solver on the modified model,
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 1b. AB, where A is 1 4 and B is 4 1c. AB, where A is 4 1 and B
The method for rating teams in Example 7.8 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, we observe whether the home team wins.
The file P07_33.xlsx contains the scores of the 1994 Big Ten basketball season. Develop ratings for the 11 teams. Which was the best team? Which was the worst team? What is the estimated home court advantage?If you are an Indiana fan, try it again after deleting Indiana’s huge loss to Minnesota.
The file P07_32.xlsx contains scores on all of the regular-season games in the NBA for the 2001–2002 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 approximately correct? (You might recall that
Discuss how the ratings change when early-season games are discounted heavily.
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 regular games of the season(in week 17), so for these final games, k
Carry out the suggestion in Modeling Issue
That is, find the ratings of the 2004 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 used in Example 7.8.
Carry out the suggestion in Modeling Issue
The file P07_28.xlsx lists the scores of all NFL games played during the 1996 season. Use this data set to rank the NFL teams from best to worst.29.. The file P07_29.xlsx lists the scores of all NFL games played during the 1997 season. Using all data except for the last game (the Super Bowl), rank
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. (These distances, along with other inputs, are in the file P07_27.xlsx.) The company can build either or both of
Modify the warehouse location model as suggested in Modeling Issue
Keep track of the changing cells and the target cell.
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 to vary from 1 to 10 in increments of
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 (see Figure 7.28 for guidance). For example, try placing the customer way up to the right, or way down to the left, or near a current
In Example 7.5, 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 1 ad might really be an average over 15 different weeks where 1
In the solution to the advertising selection model in Example 7.6, 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. Form a SolverTable to see how much
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,” that appeals primarily to women. Use the following constants
Starting with the solution to the advertising selection problem in Example 7.6, suppose the company, for whatever reason, cannot place ads on “Sports Center.”Make the appropriate changes in the model and rerun Solver. Comment on the changes to the changing cells.Then comment on the change to
As we mentioned, the advertising response function in equation (7.4) is only one of several nonlinear functions we could have used to get the same“increasing at a decreasing rate” behavior we want in Example 7.5. Another possibility is the function f (n) anb, where a and b are again constants
In judging the fit of the estimated response function in Example 7.5, we could use MAD (mean absolute deviation) instead of RMSE. MAD is the average of the absolute prediction errors.a. When you run Solver with MAD as your objective, do you get approximately the same estimated response function as
In estimating the advertising response function in Example 7.5, 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 as in the example? Try running
Continuing the previous problem (the model in part a)one step further, assume that shirts and ties are also complementary. Specifically, assume that each time a shirt is purchased (and is not accompanied by a suit purchase), 1.3 ties, on average and regardless of the price of ties, are also
Continuing Problem 7, suppose the company is selling in the United States, the United Kingdom, and Japan. Assume the unit production cost is $50, and the exchange rates are 1.56 ($/£) and 0.00821 ($/¥).Each country has its own constant elasticity demand function. The parameters for the United
In the peak-load pricing model in Example 7.4, we assumed that the capacity level is a decision variable.Assume now that capacity has already been set at 30 kwh. (Note that the cost of capacity is now a sunk cost, so it is irrelevant to the decision problem.)Change the model appropriately and run
In the peak-load pricing model in Example 7.4, the demand functions have positive and negative coefficients of prices. The negative coefficients indicate that as the price of a product increases, demand for that product decreases. The positive coefficients indicate that as the price of a product
In the complementary-product pricing model in Example 7.3, the SolverTable results in Figure 7.15 indicate that the company can sometimes increase overall profit by selling suits below cost. How far might this behavior continue? Answer by extending the SolverTable to larger values of the
In the exchange rate model in Example 7.2, 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 each country where it sells.
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 each output.)Discuss your findings.
Continuing the previous problem, create a two-way data table similar to the one-way data table in Figure 7.9. 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 profit. Explain how the values
Continuing the previous problem, calculate the mean absolute percentage error (MAPE) for each of the two fits, linear and constant elasticity, where each MAPE is the average of the absolute percentage errors for the five points. On the basis of MAPE, which curve provides the better fit?
In Example 7.1, we assumed that two points on the demand curve were given (see Figure 7.8). Suppose three additional points are estimated by Madison:(1) demand of 460 when price is $65, (2) demand of 355 when price is $75, and (3) demand of 275 when price is $85. With these new points and the
Ten different types of brownies are sold. You are thinking of developing a new brownie for sale.Brownies are rated on the basis of 5 qualities: price, chocolate flavor, chewiness, sweetness, and ease of preparation. You want to group the 10 brownies on the market into 3 clusters. Each cluster
The Wanderers Insurance Company has hired you to determine the number of sales divisions into which the country should be divided. Each division will need a president, a vice president, and a divisional staff. The time needed to call on a client will depend on the distance of the salesperson from
Suppose that you want to divide a state containing 12 cities into 5 congressional districts. How might you use IP to assign cities to districts?
Determine which products the company should produce to maximize its profit. (Hint: Use a binary changing cell for each product and a binary changing cell for each customer segment/product combination. To ensure that a customer buys only the product with the largest utility, include the following
Based on McBride and Zufryden (1988). A company is trying to determine which of five possible products to include in its product line. The fixed cost of producing each product and the unit profit for each product are listed in the file P06_98.xlsx. There are five customer segments. The number of
In the equation SEND MORE MONEY, each letter represents a different digit (0–9). The addition is done in the usual way from right to left, first adding D and E to obtain Y (possibly with a carry-over), then adding N and R, and so on. Develop a linear integer model to determine the digit
An aged merchant of Baghdad was much respected by all who knew him. He had three sons, and it was a rule of his life to treat them equally. Whenever one son received a present, the other two each received a present of equal value. One day this worthy man fell sick and died, bequeathing all of his
The following considerations also apply:■ There is an administrative cost of $5000 associated with setting up a supplier’s account. For example, if Motorola uses three suppliers, it incurs an administrative cost of $15,000.■ To ensure reliability, no supplier can supply more than 80% of
This problem is based on Motorola’s online method for choosing suppliers. Suppose Motorola solicits bids from five suppliers for eight products. The list price for each product and the quantity of each product that Motorola needs to purchase during the next year are listed in the file
The file P06_92.xlsx lists the distances between 21 U.S. cities. We want to locate liver transplant centers in a subset of these 21 cities.a. Suppose we plan to build four liver transplant centers and our goal is to minimize the maximum distance a person in any of these cities has to travel to a
We are scheduling company interviews at the annual university career fair. Five interview rooms are available.Interviews are conducted from 9 AM to 5 PM.Each company wants all its interviews conducted in a single room. The time preferences for the companies are listed in the file P06_91.xlsx.
A single-shelf bookcase can hold books that are a total of 24 feet wide. We need to store 19 books of the widths (in feet) listed in the file P06_90.xlsx.Develop a linear integer model to determine the minimum number of bookcases that are needed to store all 19 books.
Based on Kalvaitishi and Posgay (1974). A direct mail company must determine which of 10 mailing lists to use this month to generate a catalog mailing. The number of names in each list, the mainframe computer time (in hours) needed to process the list, the labor time per letter mailed for each list
Based on Angel et al. (2003). A fertilizer company is trying to determine the cheapest fertilizer mix that provides desired amounts of nutrients. The mix is made by combining the following fertilizers: SSA, SPO, GUR, TSP, KCI, FERT, and SPF. The mix cannot contain both GUR and TSP. The percentage
Powerco needs to determine a capacity expansion plan to meet Bloomington’s power needs for the next 20 years. The current capacity is 5000 kwh. The demand for the current year is 4000 kwh, and demand is expected to increase by 1000 kwh in each succeeding year. At the beginning of each year,
Consider the following puzzle. You are to select four three-letter “words” from the following list: DBA DEG ADI FFD GHI BCD FDF BAI. For each word, you earn a score equal to the position of the word’s third letter in the alphabet. For example, DBA earns a score of 1, DEG earns a score of 7,
Based on Spencer et al. (1990). When you lease 800 phone numbers from AT&T for telemarketing, AT&T uses an optimization model to tell you where you should locate calling centers to minimize your operating costs over a 10-year horizon. To illustrate the model, suppose you are considering seven
It is currently the beginning of 2006. Gotham City is trying to sell municipal bonds to support improvements in recreational facilities and highways. The face values (in thousands of dollars) of the bonds and the due dates at which principal comes due are listed in the file P06_84.xlsx. (The due
Houseco Developers is considering erecting three office buildings. The time (in years) required to complete each of them and the number of workers required to be on the job at all times are shown in the file P06_81.xlsx. After a building is completed, it brings in the following amount of rent per
Based on Muckstadt and Wilson (1968). PSI believes it will need the amounts of generating capacity (in millions of kwh) shown in the file P06_80.xlsx during the next five years. The company has a choice of building (and then operating) power plants with the capacities (in millions of kwh) and costs
This year, the company must begin and complete the five jobs shown in the file P06_79.xlsx during this 8-week period. Arthur Ross employs 4 full-time accountants who normally work 40 hours per week. If necessary, however, they can work up to 20 hours of overtime per week for which they are paid
Arthur Ross, Inc., must complete many corporate tax returns during the period February 15 to April
(Hint: Set it up so that SolverTable can solve all four problems simultaneously.)
Suppose Gotham City has n ambulance locations. Determine the locations of ambulances that maximize the number of people who live within two minutes of an ambulance.Do this separately for n 1; n 2; n 3; n
Based on Eaton et al. (1985). Gotham City has been divided into eight districts. The time (in minutes) it takes an ambulance to travel from one district to another is shown in the file P06_78.xlsx. The population of each district (in thousands) is as follows: district 1, 40; district 2, 30;
For example, costs can be expressed in units of $1 million or $100,000, and annual shipments can be expressed in units of 100,000 loaves.)
Based on Efroymson and Ray (1966). Breadco Bakeries is a new bakery chain that sells bread to customers throughout the state of Indiana. Breadco is considering building bakeries in three locations:Evansville, Indianapolis, and South Bend. Each bakery can bake up to 900,000 loaves of bread each
Based on Bean et al. (1988). Simon’s Mall has 10,000 square feet of space to rent and wants to determine the types of stores that should occupy the mall. The minimum number and maximum number of each type of store (along with the square footage of each type) are given in the file P06_74.xlsx. The
Cousin Bruzie of radio station WABC schedules radio commercials in 60-second blocks. This hour, the station has sold time for commercials of 15, 16, 20, 25, 30, 35, 40, and 50 seconds. Determine the minimum number of 60-second blocks of commercials that must be scheduled to fit in all the current
Determine whether there is an arrangement of songs satisfying these restrictions. (Hint: You do not need a target cell when using Solver. In the Solver dialog box, just leave the Target Cell box empty.)
You have been assigned to arrange the songs on the cassette version of Madonna’s latest album. (Feel free to substitute your own favorite rock star for Madonna!) A cassette tape has two sides (1 and 2).The songs on each side of the cassette must total between 14 and 16 minutes in length. The
Based on Fitzsimmons and Allen (1983). The State of Texas frequently audits companies doing business in Texas. Because these companies often have headquarters located outside the state, auditors must be sent to out-of-state locations. Each year, auditors must make 500 trips to cities in the
Based on Zangwill (1992). Hallco runs a day shift and a night shift. Regardless of the number of units produced, the only production cost during a shift is a setup cost. It costs $8000 to run the day shift and$4500 to run the night shift. Demand for the next two days is as follows: day 1, 2000;
Based on Boykin (1985). Chemco annually produces 359 million pounds of the chemical maleic anhydride.A total of four reactors are available to produce maleic anhydride. Each reactor can be run on one of three settings.The cost (in thousands of dollars) and pounds produced (in millions) annually for
Based on Westerberg, Bjorklund, and Hultman (1977).Newcor’s steel mill has received an order for 25 tons of steel. The steel must be 5% carbon and 5% molybdenum by weight. The steel is manufactured by combining three types of metal: steel ingots, scrap steel, and alloys. Four individual steel
Based on Liggett (1973). A court decision has stated that the enrollment of each high school in Metropolis must be at least 20% black. The numbers of black students and white students in each of the city’s five school districts are listed in the file P06_67.xlsx. The distance (in miles) that a
Eastinghouse ships 12,000 capacitors per month to its customers. The capacitors can be produced at three different plants. The production capacity, fixed monthly cost of operation, and variable cost of producing a capacitor at each plant are given in the file P06_66.xlsx. The fixed cost for a plant
The minimum order from a vendor is 200 computers. Determine how to minimize the cost of purchasing the needed computers.
State University must purchase 1100 computers from three vendors. Vendor 1 charges $500 per computer plus a total delivery charge of $5000. Vendor 2 charges$350 per computer plus a total delivery charge of$4000. Vendor 3 charges $250 per computer plus a total delivery charge of $6000. Vendor 1 will
Suppose in the previous problem that each customer’s demand must be met from a single warehouse. Solve the problem with this restriction.
Heinsco produces tomato sauce at five different plants.The tomato sauce is then shipped to one of three warehouses, where it is stored until it is shipped to one of the company’s four customers. The following inputs for the problem are given in the file P06_61.xlsx:■ The plant capacities (in
Ford has four automobile plants. Each is capable of producing the Focus, Mustang, or Crown Victoria, but it can produce only one of these cars. The fixed cost of operating each plant for a year and the variable cost of producing a car of each type at each plant are given in the file P06_59.xlsx.
During each period when production occurs, a setup cost of $250 and a per-unit production cost of $2 are incurred. At the end of each period, a per-unit holding cost of $1 is incurred. Determine the cost-minimizing production schedule.
During the next five periods, the demands listed in the file P06_58.xlsx must be met on time. At the beginning of period 1, the inventory level is
Eastinghouse sells air conditioners. The annual demand for air conditioners in each region of the country is as follows: East, 100,000; South, 150,000; Midwest, 110,000; and West, 90,000. Eastinghouse is considering building its air conditioners in four different cities:New York, Atlanta, Chicago,
Comquat owns four production plants at which computer workstations are produced. Comquat can sell up to 20,000 computers per year at a price of $3500 per computer. For each plant, the production capacity, the production cost per computer, and the fixed cost of operating a plant for a year are given
Bookco Publishers is considering publishing five textbooks.The maximum number of copies of each textbook that can be sold, the variable cost of producing each textbook, the selling price of each textbook, and the fixed cost of a production run for each book are given in the file P06_55.xlsx. For
Find Pigskin’s optimal production policy if, in addition to the given production and holding costs, there is a fixed cost of $5000 during any month in which there is positive production. Assume now that storage capacity is 200 (hundreds of) footballs.
Consider the Pigskin example (Example 3.3) from Chapter
Fruit Computer produces two types of computers: Pear computers and Apricot computers. The relevant data are given in the file P06_52.xlsx. The equipment cost is a fixed cost that is incurred if any of this type of computer is produced. A total of 3000 chips and 1200 hours of labor are available.a.
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