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practical management science
Practical Management Science, Revised 3rd Edition Wayne L Winston, S. Christian Albright - Solutions
The lead time for her supplier to deliver an order is L working days, where there are 6 working days in a week. (Essentially, you can ignore Sundays.) The weekly demand is for 20 tape measures.What reorder point should the manager use if L 3; if L 5; if L 10? (Hint: The manager should plan
The manager of a hardware store decides to use the EOQ with shortages model to determine the ordering policy for tape measures. Using economic considerations, the manager determines that she should use an Cost of goods sold per year Average value of on-hand inventory order quantity of Q 30
The Gilette Company buys a product using the price schedule given in the file P13_32.xlsx. The company estimates the unit holding cost at 10% of the purchase price and the ordering cost at $40 per order. Gilette’s annual demand is 460 units.a. Determine how often the company should order.b.
Each year, Shopalot Stores sells 10,000 cases of soda.The company is trying to determine how many cases to order each time it orders. It costs $50 to process each order, and the cost of carrying a case of soda in inventory for 1 year is 20% of the purchase price. The soda supplier offers Shopalot
A consulting firm is trying to determine how to minimize the annual costs associated with purchasing highquality paper for its printers. Each time an order is placed, an ordering cost of $20 is incurred. The price per ream of printer paper depends on Q, the number of reams ordered, as shown in the
The efficiency of an inventory system is often measured by the turnover ratio. The turnover ratio (TR) is defined by TR a. Does a high turnover ratio indicate an efficient inventory system?b. If the EOQ model is being used, determine TR in terms of K, D, h, and Q.c. Suppose that D increases. Show
Based on Baumol (1952). Money in your savings account earns interest at a 10% annual rate. Each time you go to the bank, you waste 15 minutes in line, and your time is worth $10 per hour. During each year, you need to withdraw $10,000 to pay your bills.a. How often should you go to the bank?b. Each
Consider the basic EOQ model.We want to know the sensitivity of (1) the optimal order quantity, (2) the sum of the annual order cost and the annual holding cost (not including the annual purchase cost cD), and(3) the time between orders to various parameters of the problem.a. How do (1), (2), and
A bakery that orders cartons of bread mix has used an EOQ model to determine that an order quantity of 90 cartons per order is economically optimal. The bakery needs 150 cartons per month to meet demand. It takes L days for the bakery’s supplier to deliver an order.When should the bakery place
Each month, a gas station sells 4000 gallons of gasoline.Each time the parent company refills the station’s tanks, it charges the station $100 plus $2.40 per gallon.The annual cost of holding a gallon of gasoline is $0.30.a. How large should the station’s orders be?b. How many orders per year
The problem in Example 13.9 assumes that the heaviest demand occurs in the second (post-April) phase of selling. It also assumes that capacity is higher in the second production opportunity than in the first.Suppose the situation is reversed, so that the higher capacity and most of the demand occur
Change the ordering simulation so that emergency orders are never made. Instead, assume that all excess demand is backlogged. Now the inventory position is the amount on hand, plus the amount on order, minus the backlog. Simulate the same (s, S) policies as in the example.
Change the ordering simulation so that emergency orders are never made. If demand in any week is greater than supply, the excess demand is simply lost. Simulate the same (s, S) policies as in the example.Skill-Extending Problem
Change the ordering simulation so that the lead time can be 1, 2, 3, or 4 weeks with probabilities 0.5, 0.2, 0.2, and 0.1, respectively. Also, assume that based on previous orders, orders of sizes 350, 0, and 400 are scheduled to arrive at the beginnings of weeks 2, 3, and 4, respectively. Simulate
What (R,Q) policy should it use? Then find the model 3 cost parameter (the cost per cycle with a shortage) that is equivalent to this service level.
Turn the previous problem around. Now assume that the store’s service level requirement obligates it to meet customer demand on 99% of all order cycles. In other words, use model
The first (R,Q) model in this section assumes that the total shortage cost is proportional to the amount of demand that cannot be met from on-hand inventory.Similarly, the second model assumes that the service level constraint is in terms of the fill rate, the fraction of all customer demand that
We claimed that the critical fractile formula, equation(13.8), is appropriate because the optimal Q should satisfy cunder (1 F(Q)) cover F(Q), that is, the cost of understocking times the probability of understocking should equal the cost of overstocking times the probability of overstocking.
In Example 13.7, we discussed the equivalence between the model with shortage costs and the model with a service level constraint. We also showed how to see this equivalence with SolverTable. Extend the SolverTable in the Ordering Cameras 1.xlsx file, with the unit shortage cost as the single input
In both (R,Q) models, the one with a shortage cost and the one with a service level constraint, we set up Solver so that the multiple k is constrained to be nonnegative.The effect is that the reorder point R will be no less than the mean demand during lead time, and the expected safety stock will
Change the model in the file Ordering Cameras 2.xlsx slightly to allow a random lead time with a given mean and standard deviation. If the mean lead time is 2 weeks, and the standard deviation of lead time is half a week, find the optimal solution if the company desires a fill rate of 98.5%.
In the first (R,Q) model in Example 13.7, the one with a shortage cost, we let both Q and the multiple k be changing cells. However, we stated that the optimal Q depends mainly on the fixed ordering cost, the holding cost, and the expected annual demand. This implies that a good approximation to
We saw in Example 13.6 that the optimal order quantities with the triangular and normal demand distributions are very similar (171 versus 174). Perhaps this is because these two distributions, with the parameters used in the example, have similar shapes. Explore whether this similarity in optimal
As stated in Example 13.6, the critical fractile analysis is useful for finding the optimal order quantity, but it doesn’t (at least by itself) show the probability distribution of net profit. Use @RISK, as in Chapter 11, to explore this distribution. Actually, do it twice, once with the
Consider each change to the monetary inputs (the purchase cost, the selling price, and the salvage price) one at a time in Example 13.6. For each such change, either up or down, describe how the cost of understocking and the cost of overstocking change, how the critical fractile changes, and how
What region is the optimal ordering quantity in if there is no price break at all (k 0). How do you reconcile this with your SolverTable findings?
In the quantity discount model in Example 13.2, the minimum total annual cost is region 3 is clearly the best. Evidently, the larger unit purchase costs in the other two regions make these two regions unattractive.When would a switch take place? To answer this question, change the model slightly.
In the basic EOQ model, revenue is often omitted from the model. The reasoning is that all demand will be sold at the given selling price, so revenue is a fixed quantity that is independent of the order quantity.Change that assumption as follows. Make selling price a decision variable, which must
In the basic EOQ model in Example 13.1, suppose that the fixed cost of ordering and the unit purchasing cost are both multiplied by the same factorf. Use SolverTable to see what happens to the optimal order quantity and the corresponding annual fixed order cost and annual holding cost as f varies
Modify the synchronized ordering model in Example 13.5 slightly so that you can use a two-way SolverTable on the fixed costs. Specifically, enter a formula in cell B9 so that the fixed cost of ordering kings alone is equal to the fixed cost of ordering queens alone. Then let the two inputs for
In Example 13.4, we showed why a company might invest to reduce its setup cost. It all depends on how much this investment costs, as specified (in the model)by the cost of a 10% reduction in the setup cost. Use SolverTable to see how the results change as this cost of a 10% reduction varies. You
In Example 13.3, we used SolverTable to show what happens when the unit shortage cost varies. As the table indicates, the company orders more and allows more backlogging as the unit shortage cost decreases.Redo the SolverTable, this time trying even smaller unit shortage costs. Explain what happens
The quantity discount model in Example 13.2 uses one of two possible types of discount structures. It assumes that if the company orders 600 units, say, each unit costs $28. This provides a big incentive to jump up to a higher order quantity. For example, the total purchasing cost of 499 units is
In the quantity discount model in Example 13.2, the minimum total annual cost in region 3 is clearly the best. Evidently, the larger unit purchase costs in the other two regions make these two regions unattractive.Could region 1 ever be best? What about region 2? To answer these questions, assume
In the quantity discount model in Example 13.2, suppose we want to see how the optimal order quantity and the total annual cost vary as the fixed cost of ordering varies. Use a two-way SolverTable to perform this analysis, allowing the fixed cost of ordering to vary from $50 to $200 in increments
If the lead time in Example 13.1 changes from one week to two weeks, how is the optimal policy affected?Does the optimal order quantity change?
In the basic EOQ model in Example 13.1, suppose that the fixed cost of ordering is $500. Use Solver to find the new optimal order quantity. How does it compare to the optimal order quantity in the example? Could you have predicted this from equation (13.4)?
Suppose you are using an underwater probe to search for a sunken ship. At any time in the search, your probe is located at some point (x,y) in a grid, where the distance between lines in the grid is some convenient unit such as 100 meters. The sunken ship is at some unknown location on the grid,
Similarly, the way the uncertainty about the units sold per pig in 2007 is stated suggests using the General distribution from Section 11.3 in Chapter 11.)
Nucleon is trying to determine whether to produce a new drug that makes pigs healthier. The product will be sold in the years 2007 to 2011. The following information is relevant:■ A fixed cost is incurred on 1/1/2006 and will be between$1 billion and $5 billion. There is a 20%chance the fixed
Similarly, the way the uncertainty about the annual growth rate is stated suggests using the Cumul distribution from Section 11.3.)
It is January 1, 2006, and Merck is trying to determine whether to continue development of a new drug. The following information is relevant. You can assume that all cash flows occur at the ends of the respective years.■ Clinical trials (the trials where the drug is tested on humans) are equally
It is January 1, 2006, and Lilly is considering developing a new drug called Dialis. We are given the following information■ On March 15, 2006, Lilly incurs a fixed cost that is assumed to follow a triangular distribution with best case $10 million, most likely case $35 million, and worst case
It is now May 1, 2006, and GM is deciding whether to produce a new car. The following information is relevant.■ The fixed cost of developing the car is incurred on January 1, 2007 and is assumed to follow a triangular distribution with smallest possible cost $300 million, most likely cost $400
You win.Note that only 4 tosses need to be generated for the house, but more tosses might need to be generated for you, depending on your strategy. Develop a simulation and run it for at least 1000 iterations for each of the strategies listed previously. For each strategy, what are the two values
The house tosses a 3 and then a
You win.■ You toss a 3 and then a 4 for total of
You lose.■ You toss a 6 and stop. The house tosses a 3 and then a
You lose because a tie goes to the house.■ You toss a 3 and then a
In this version of “dice blackjack,” you toss a single die repeatedly and add up the sum of your dice tosses.Your goal is to come as close as possible to a total of 7 without going over. You may stop at any time. If your total is 8 or more, you lose. If your total is 7 or less, the “house”
Based on Bukiet et al. (1997). Many Major League teams (including Oakland, Boston, LA Dodgers, and Toronto) use mathematical models to evaluate baseball players. A common measure of a player’s offensive effectiveness is the number of runs generated per inning(RPI) if a team were made up of nine
You are unemployed, 21 years old, and searching for a job. Until you accept a job offer, the following situation occurs. At the beginning of each year, you receive a job offer. The annual salary associated with the job offer is equally likely to be any number between$20,000 and $100,000. You must
The Tinkan Company produces 1-pound cans for the Canadian salmon industry. Each year, the salmon spawn during a 24-hour period and must be canned immediately. Tinkan has the following agreement with the salmon industry. The company can deliver as many cans as it chooses. Then the salmon are caught.
Chemcon has taken over the production of Nasacure from a rival drug company. Chemcon must build a plant to produce Nasacure by the beginning of 2007.After the plant is built, the plant’s capacity cannot be changed. Each unit sold brings in $10 in revenue. The fixed cost (in dollars) of producing
Based on Hoppensteadt and Peskin (1992). The following model (the Reed–Frost model) is often used to model the spread of an infectious disease. Suppose that at the beginning of period 1, the population consists of 5 diseased people (called infectives) and 95 healthy people (called susceptibles).
Rework the previous problem for a case in which the 1-year warranty requires you to pay for the new device even if failure occurs during the warranty period.Specifically, if the device fails at time t, measured relative to the time it went into use, you must pay $100t for a new device. For example,
Suppose you buy an electronic device that you operate continuously. The device costs you $100 and carries a 1-year warranty. The warranty states that if the device fails during its first year of use, you get a new device for no cost, and this new device carries exactly the same warranty. However,
Truckco produces the OffRoad truck. The company wants to gain information about the discounted profits earned during the next three years. During a given year, the total number of trucks sold in the United States is 500,000 50,000G 40,000I where G is the percentage increase in gross domestic
Dord Motors is considering whether to introduce a new model called the Racer. The profitability of the Racer will depend on the following factors:■ The fixed cost of developing the Racer is equally likely to be $3 or $5 billion.■ Year 1 sales are normally distributed with mean 200,000 and
Toys For U is developing a new Hannah Montana doll.The company has made the following assumptions:■ It is equally likely that the doll will sell for 2, 4, 6, 8, or 10 years.■ At the beginning of year 1, the potential market for the doll is 1 million. The potential market grows by an average of
So the future cash flows after abandonment should disappear.)
Estimate the mean and standard deviation of the project with the abandonment option.How much would you pay for the abandonment option? (Hint: You can abandon a project at most once. Thus in year 5, for example, you abandon only if the sum of future expected NPVs is less than the year 5 abandonment
You are considering a 10-year investment project. At present, the expected cash flow each year is $1000.Suppose, however, that each year’s cash flow is normally distributed with mean equal to last year’s actual cash flow and standard deviation $100. For example, suppose that the actual cash
Mary Higgins is a freelance writer with enough spare time on her hands to play the stock market fairly seriously.Each morning, she observes the change in stock price of a particular stock and decides whether to buy or sell, and if so, how many shares to buy or sell. We assume that on day 1, she has
You want to estimate your annual return over a 50-year period. If you end with F dollars, then your annual return is (F/500)1/501. For example, if you end with $10,000, your annual return is 201/50 1 0.062, or 6.2%. Run 1000 replications of an appropriate simulation. Based on the results, you
You begin year 1 with $500. At the beginning of each year, you put half of your money under your mattress and invest the other half in Whitewater stock. During each year, there is a 50% chance that the Whitewater stock will double, and there is a 50% chance that you will lose half of your
Consider an oil company that bids for the rights to drill in offshore areas. The value of the right to drill in a given offshore area is highly uncertain, as are the bids of the competitors. This problem demonstrates the “winner’s curse.” The winner’s curse states that the optimal bidding
Amanda has 30 years to save for her retirement. At the beginning of each year, she puts $5000 into her retirement account. At any point in time, all of Amanda’s retirement funds are tied up in the stock market.Suppose the annual return on stocks follows a normal distribution with mean 12% and
Based on Kelly (1956). You currently have $100. Each week, you can invest any amount of money you currently have in a risky investment. With probability 0.4, the amount you invest is tripled (e.g., if you invest$100, you increase your asset position by $300), and, with probability 0.6, the amount
We are trying to determine the proper capacity level for a new electric car. A unit of capacity gives us the potential to produce one car per year. It costs $10,000 to build a unit of capacity, and the cost is charged equally over the next 5 years. It also costs $400 per year to maintain a unit of
The annual demand for Prizdol, a prescription drug manufactured and marketed by the NuFeel Company, is normally distributed with mean 50,000 and standard deviation 12,000. We assume that demand during each of the next 10 years is an independent random draw from this distribution. NuFeel needs to
Freezco sells refrigerators. Any refrigerator that fails before it is three years old is replaced free. Of all refrigerators, 3% fail during their first year of operation;5% of all one-year-old refrigerators fail during their second year of operation; and 7% of all two-year-old refrigerators fail
Consider a drill press containing three drill bits. The current policy (called individual replacement) is to replace a drill bit when it fails. The firm is considering changing to a block replacement policy in which all three drill bits are replaced whenever a single drill bit fails. Each time the
Consider a device that requires two batteries to function.If either of these batteries dies, the device will not work. Currently, two brand new batteries are in the device, and we have three extra brand new batteries.Each battery, after it is placed in the device, lasts a random amount of time that
You have been asked to simulate the cash inflows to a toy company for the next year. Monthly sales are independent random variables. Mean sales for the months January to March and October to December are$80,000, and mean sales for the months April to September are $120,000. The standard deviation
Based on Marcus (1990). The Balboa mutual fund has beaten the Standard and Poor’s 500 during 11 of the last 13 years. People use this as an argument that you can beat the market. Here is another way to look at it that shows that Balboa’s beating the market 11 out of 13 times is not unusual.
A ticket from Indianapolis to Orlando on Deleast Airlines sells for $150. The plane can hold 100 people. It costs Deleast $8000 to fly an empty plane. Each person on the plane incurs variable costs of $30 (for food and fuel). If the flight is overbooked, anyone who cannot get a seat receives $300
Suppose you have invested 25% of your portfolio in four different stocks. The mean and standard deviation of the annual return on each stock are as shown in the file P12_37.xlsx. The correlations between the annual returns on the four stocks are also shown in this file.a. What is the probability
You now have $1000, all of which is invested in a sports team. Each year there is a 60% chance that the value of the team will increase by 60% and a 40%chance that the value of the team will decrease by 60%. Estimate the mean and median value of your investment after 100 years. Explain the large
You now have $3.You will toss a fair coin four times.Before each toss you can bet any amount of your money(including none) on the outcome of the toss. If heads comes up, you win the amount you bet. If tails comes up, you lose the amount you bet.Your goal is to reach $6. It turns out that you can
Your company assembles two components into a finished product. You get component 1 from one supplier and component 2 from another supplier. You just received an order for the product, so you immediately request a component 1 and a component 2 from the suppliers. The shipping times for the suppliers
You are playing Andy Roddick in tennis, and you have a 42% chance of winning each point. (You are good!)a. Use simulation to estimate the probability you will win a particular game. Note that the first player to score at least 4 points and have at least 2 more points than his opponent wins the
Based on Morrison and Wheat (1984). When his team is behind late in the game, a hockey coach usually waits until there is one minute left before pulling the goalie. Actually, coaches should pull their goalies much sooner. Suppose that if both teams are at full strength, each team scores an average
A martingale betting strategy works as follows. We begin with a certain amount of money and repeatedly play a game in which we have a 40% chance of winning any bet. In the first game, we bet $1. From then on, every time we win a bet, we bet $1 the next time.Each time we lose, we double our previous
The game of Chuck-a-Luck is played as follows: You pick a number between 1 and 6 and toss three dice. If your number does not appear, you lose $1. If your number appears x times, you win $x. On the average, how much money will you win or lose on each play of the game? Use simulation to find out.
The Mutron Company is thinking of marketing a new drug used to make pigs healthier. At the beginning of the current year, there are 1,000,000 pigs that might use the product. Each pig will use Mutron’s drug or a competitor’s drug once a year. The number of pigs is forecasted to grow by an
Suppose that GLC earns a $4000 profit each time a person buys a car.We want to determine how the expected profit earned from a customer depends on the quality of GLC’s cars.We assume a typical customer will purchase 10 cars during her lifetime. She will purchase a car now(year 1) and then
Based on Babich (1992). Suppose that each week, each of 300 families buys a gallon of orange juice from company A, B, or C. Let pA denote the probability that a gallon produced by company A is of unsatisfactory quality, and define pB and pC similarly for companies B and C. If the last gallon of
Seas Beginning sells clothing by mail order. An important question is when to strike a customer from their mailing list. At present, they strike a customer from their mailing list if a customer fails to order from six consecutive catalogs. They want to know whether striking a customer from their
Suppose that Coke and Pepsi are fighting for the cola market. Each week, each person in the market buys one case of Coke or Pepsi. If the person’s last purchase was Coke, there is a 0.90 probability that this person’s next purchase will be Coke; otherwise, it will be Pepsi. (We are considering
The investor uses the following strategy.At the end of March, he exercises the option only if the stock price is above $51.50. At the end of April, he exercises the option (assuming he hasn’t exercised it yet) only if the price is above $50.75. At the end of May, he exercises the option (assuming
The contract allows him to buy 100 shares of ABC stock at the end of March, April, or May at a guaranteed price of $50 per share. He can exercise this option at most once. For example, if he purchases the stock at the end of March, he cannot purchase more in April or May at the guaranteed price. If
Suppose an investor has the opportunity to buy the following contract (a stock call option) on March
A knockout call option loses all value at the instant the price of the stock drops below a given “knockout level.” Determine a fair price for a knockout call option when the current stock price is $20, the exercise price is $21, the knockout price is $19.50, the mean annual growth rate of the
Cryco stock currently sells for $69. The annual growth rate of the stock is 15%, and the stock’s annual volatility is 35%. The risk-free rate is currently 5%. You have bought a 6-month European put option on this stock with an exercise price of $70.a. Use @RISK to value this option.b. Use @RISK
For the data in the previous problem, the following is an example of a butterfly spread: sell two calls with an exercise price of $50, buy one call with an exercise price of $40, and buy one call with an exercise price of$60. Simulate the cash flows from this portfolio.
If you own a stock, buying a put option on the stock will greatly reduce your risk. This is the idea behind portfolio insurance. To illustrate, consider a stock(Trumpco) that currently sells for $56 and has an annual volatility of 30%. Assume the risk-free rate is 8%, and you estimate that the
Each year after year 5, the company examines sales. If fewer than 90,000 cars were sold that year, there is a 50% chance the car won’t be sold after that year. Modify the model and run the simulation. Keep track of two outputs: NPV(through year 10) and the number of years of sales.
However, the car might sell through year
Change the new car simulation from Example 12.5 as follows. It is the same as before for years 1 to 5, including depreciation through year
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