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operations research an introduction
Operations Research Applications And Algorithms 4th Edition Wayne L. Winston - Solutions
10 Payoff Insurance Company charges a customer according to his or her accident history. A customer who has had no accident during the last two years is charged a$100 annual premium. Any customer who has had an accident during each of the last two years is charged a $400 annual premium. A customer
8 Three balls are divided between two containers. During each period a ball is randomly chosen and switched to the other container.a Find (in the steady state) the fraction of the time that a container will contain 0, 1, 2, or 3 balls.b If container 1 contains no balls, on the average how many
7 Consider two stocks. Stock 1 always sells for $10 or$20. If stock 1 is selling for $10 today, there is a .80 chance that it will sell for $10 tomorrow. If it is selling for $20 today, there is a .90 chance that it will sell for $20 tomorrow.Stock 2 always sells for $10 or $25. If stock 2 sells
6 This problem will show why steady-state probabilities are sometimes referred to as stationary probabilities. Let p1, p2, . . . , ps be the steady-state probabilities for an ergodic chain with transition matrix P. Also suppose that with probability pi, the Markov chain begins in state i.a What is
5 A square matrix is said to be doubly stochastic if its entries are all nonnegative and the entries in each row and each column sum to 1. For any ergodic, doubly stochastic matrix, show that all states have the same steady-state probability.
4 At the beginning of each year, my car is in good, fair, or broken-down condition. A good car will be good at the beginning of next year with probability .85; fair with probability .10; or broken-down with probability .05. A fair car will be fair at the beginning of the next year with probability
2 For the gambler’s ruin problem (Example 1), why is it unreasonable to talk about steady-state probabilities?
1 Find the steady-state probabilities for Problem 1 of Section 17.3.
With the assumption that each Monopoly player who goes to Jail stays until he or she rolls doubles or has spent three turns in Jail, the steady-state probability of a player landing on any Monopoly square has been determined by Ash and Bishop (1972) (see Table 3).†These steady-state probabilities
In Example 4, suppose that each customer makes one purchase of cola during any week(52 weeks = 1 year). Suppose there are 100 million cola customers. One selling unit of cola costs the company $1 to produce and is sold for $2. For $500 million per year, an advertising firm guarantees to decrease
6 Which of the following chains is ergodic?
5 Fifty-four players (including Gabe Kaplan and James Garner) participated in the 1980 World Series of Poker.Each player began with $10,000. Play continued until one player had won everybody else’s money. If the World Series of Poker were to be modeled as a Markov chain, how many absorbing states
2 Is the Markov chain of Section 17.3, Problem 1, an ergodic Markov chain?
1 In Example 1, what is the period of states 1 and 3?
3 In Example 2, determine the following n-step transition probabilities:a After two balls are painted, what is the probability that the state is [0 2 0]?b After three balls are painted, what is the probability that the state is [0 1 1]? (Draw a diagram like Figure 4.)
2 The following questions refer to Example 1.a After playing the game twice, what is the probability that I will have $3? How about $2?b After playing the game three times, what is the probability that I will have $2?
1 Each American family is classified as living in an urban, rural, or suburban location. During a given year, 15% of all urban families move to a suburban location, and 5% move to a rural location; also, 6% of all suburban families move to an urban location, and 4% move to a rural location;finally,
Suppose the entire cola industry produces only two colas. Given that a person last purchased cola 1, there is a 90% chance that her next purchase will be cola 1. Given that a person last purchased cola 2, there is an 80% chance that her next purchase will be cola 2.1 If a person is currently a cola
6 In Problem 3, suppose a machine that breaks down returns to service three days later (for instance, a machine that breaks down during day 3 would be back in working order at the beginning of day 6). Determine a transition probability matrix for this situation
5 Let Xt be the location of your token on the Monopoly board after t dice rolls. Can Xt be modeled as a Markov chain? If not, how can we modify the definition of the state at time t so that X0, X1, . . . , Xt, . . . would be a Markov chain? (Hint: How does a player go to Jail? In this problem,
4 Referring to Problem 1, suppose that tomorrow’s Smalltown weather depends on the last two days of Smalltown weather, as follows: (1) If the last two days have been sunny, then 95% of the time, tomorrow will be sunny.(2) If yesterday was cloudy and today is sunny, then 70% of the time, tomorrow
3 A company has two machines. During any day, each machine that is working at the beginning of the day has a 1 3chance of breaking down. If a machine breaks down during the day, it is sent to a repair facility and will be working two days after it breaks down. (Thus, if a machine breaks down
2 Consider an inventory system in which the sequence of events during each period is as follows. (1) We observe the inventory level (call it i) at the beginning of the period.(2) If i 1, 4 i units are ordered. If i 2, 0 units are ordered. Delivery of all ordered units is immediate. (3) With
1 In Smalltown, 90% of all sunny days are followed by sunny days, and 80% of all cloudy days are followed by cloudy days. Use this information to model Smalltown’s weather as a Markov chain.
In recent years, students of finance have devoted much effort to answering the question of whether the daily price of a stock share can be described by a Markov chain. Suppose the daily price of a stock share (such as CSL Computer stock) can be described by a Markov chain. What does that tell us?
Choosing Balls Find the transition matrix for Example 2.
Find the transition matrix for Example 1.
Let X0 be the price of a share of CSL Computer stock at the beginning of the current trading day. Also, let Xt be the price of a share of CSL stock at the beginning of the tth trading day in the future. Clearly, knowing the values of X0, X1, . . . , Xt tells us something about the probability
An urn contains two unpainted balls at present. We choose a ball at random and flip a coin. If the chosen ball is unpainted and the coin comes up heads, we paint the chosen unpainted ball red; if the chosen ball is unpainted and the coin comes up tails, we paint the chosen unpainted ball black. If
At time 0, I have $2. At times 1, 2, . . . , I play a game in which I bet $1. With probability p, I win the game, and with probability 1 p, I lose the game. My goal is to increase my capital to $4, and as soon as I do, the game is over. The game is also over if my capital is reduced to $0. If we
7 Use LINGO to determine the values of q and r that minimize expected annual cost for Example 5. How close are your answers to those given in the text?
6 A company currently has two warehouses. Each warehouse services half the company’s demand, and the annual demand serviced by each warehouse is N(10,000, 1,000,000). The lead time for meeting demand is 1/10 year.The company wants to meet 95% of all demand on time.Assume that the EOQ at each
5 A fur dealer buys fur coats for $100 each and sells them for $200 each. He believes that the demand for coats is N(100, 100). Any coat not sold can be sold to a discount house for $100, but the fur dealer believes he must charge himself a cost of 10¢ per dollar invested in a fur coat that is
3 We are given the following information about a product:Cost of placing an order $100 Cost per item $5 Sale price per item $8 Annual holding cost 40% of cost of item Annual demand 5,000 units Lead time demand N(20, 900)a If the reorder point that minimizes expected cost is 80, what is
2 An optometrist orders eyeglass frames at a cost of $40 per frame and sells each frame for $70. Annual holding cost is 20% of the optometrist’s cost of purchasing a frame. Each time frames are ordered, a cost of $200 is incurred. Because of lost goodwill, a cost of $50 is incurred each time a
3 Consider the exchange surface in Figure 11. The current inventory policy has yielded 3,586 orders per year, an AII of $1,367,000, and 0.89% shortages.a Without changing orders per year and AII, by how much can shortages be improved?b If AII and shortages are maintained at current levels, by how
2 Explain how to draw an exchange curve where the xcoordinate is AII and the y-coordinate is percentage of all requests for stock that result in shortages.
A company stocks two products. Relevant information is given in Table 17.1 Draw an exchange curve.2 Currently, the company is ordering each product ten times per year. Use the exchange curve to demonstrate to management that this is an unsatisfactory ordering policy.3 Suppose that management limits
3 Suppose we have found the optimal (R, S) policy for the back-ordered case and that S 50. Is the following true or false?The optimal S for the lost sales case has S > 50.
2 Chicago’s Treadway Tires Dealer must order tires from its national warehouse. It costs $10,000 to place an order and $400 to review the inventory level. Annual tire sales are N(20,000, 4,000,000). It costs $10 per year to hold a tire in inventory, and each order arrives two weeks after being
1 A hospital must order the drug Porapill from Daisy Drug Company. It costs $500 to place an order and $30 to review the hospital’s inventory of the drug. Annual demand for the drug is N(10,000, 640,000), and it costs $5 to hold one unit in inventory for one year. Orders arrive one month after
Lowland Appliance replenishes its stock of color TVs three times a year. Each order takes1/9 year to arrive. Annual demand for color TVs is N(990, 1,600). The cost of holding one color TV in inventory for one year is $100. Assume that all shortages are backlogged, with a shortage cost of $150 per
4 A firm experiences demand with a mean of 100 units per day. Lead time demand is normally distributed, with a mean of 1,000 units and a standard deviation of 200 units. It costs$6 to hold one unit for one year. If the firm wants to meet 90% of all demand on time, what will be the annual cost of
2 For Problem 2 of Section 16.6, determine the reorder point that yields 80%, 90%, 95%, and 99% values of SLM1.What reorder point would yield an average of two stockouts per year?
1 For Problem 1 of Section 16.6, determine the reorder point that yields 80%, 90%, 95%, and 99% values of SLM1.What reorder point would yield an average of 0.5 stockout per year?
Bads, Inc., sells an average of 1,000 food processors each year. Each order for food processors placed by Bads costs $50. The lead time is one month. It costs $10 to hold a food processor in inventory for one year. Annual demand for food processors is normally distributed, with a standard deviation
Suppose that for a given inventory situation, average annual demand is 1,000 and the EOQ is 100. Demand during a lead time is random and is described by the probability distribution in Table 12. For a reorder point of 30 units, determine SLM1 and SLM2.
9 Suppose annual demand for a product is normally distributed, with a mean of 600 and a variance of 300.Suppose that the lead time for an order is always one month.Show (without using Equation (8)) that the lead time demand has mean 50, variance 25, and standard deviation 5. Assume that the demands
8 Explain the following statement: Faster-moving items require larger safety stocks than slower-moving items. (Hint:Does E(D)/q large imply that an item is fast-moving or slowmoving?)
7 Suppose that a cost of S dollars (independent of the size of the stockout) is incurred whenever a stockout occurs during a cycle. Under the assumption of backlogged demand, use marginal analysis to determine the reorder point.
5 In Figure 3, assume that demand occurs at a constant rate during each cycle. Approximate the average level of onhand inventory between t 0 and t 7. Also approximate the average number of shortages. Does the assumption that the average shortage level is small relative to the average level of
3 We are given the following information for a product:Order cost $50 Annual demand N(960, 3,072.49)Annual holding cost $6/item/year Shortage cost $80 per unit Lead time one month Sales price $40 per unit Product cost $30 per unit a Determine the order quantity and the reorder point
2 Furnco sells secretarial chairs. Annual demand is normally distributed, with mean of 1,040 chairs and standard deviation of 50.99 chairs. Furnco orders its chairs from its flagship store. It costs $100 to place an order, and the lead time is two weeks. Furnco estimates that each stockout causes a
1 A hospital orders its blood from a regional blood bank.Each year, the hospital uses an average of 1,040 pints of Type O blood. Each order placed with the regional blood bank incurs a cost of $20. The lead time for each order is one week. It costs the hospital $20 to hold 1 pint of blood in
Each year, a computer store sells an average of 1,000 boxes of disks. Annual demand for boxes of disks is normally distributed with a standard deviation of 40.8 boxes. The store orders disks from a regional distributor. Each order is filled in two weeks. The cost of placing each order is $50, and
2 Assume that the Federal Reserve Board can control the growth rate of the U.S. money supply. Also assume that during a year in which the money supply grows by x%, the Gross Domestic Product (GDP) grows by Zx%, where Z is a known random variable. The government has decided it wants the GDP to grow
Condo Construction Company is bidding on an important construction job. The job will cost $2 million to complete. One other company is bidding for the job. Condo believes that the opponent’s bid is equally likely to be any amount between $2 million and $4 million.If Condo wants to maximize
6 I. L. Pea is a well-known mail-order company. During the Christmas rush (from November 1 to December 15), the number of orders that I. L. Pea must fill each day (five days per week) is normally distributed, with a mean of 2,000 and a standard deviation of 500. I. L. Pea must determine how many
5 Motorama TV estimates the annual demand for its TVs is (and will be in the future) normally distributed, with a mean of 6,000 and standard deviation of 2,000. Motorama must determine how much production capacity it should have. The cost of building enough production capacity to make 1,000 sets
4 A hot dog vendor at Wrigley Field sells hot dogs for$1.50 each. He buys them for $1.20 each. All the hot dogs he fails to sell at Wrigley Field during the afternoon can be sold that evening at Comiskey Park for $1 each. The daily demand for hot dogs at Wrigley Field is normally distributed with a
3 Joe is selling Christmas trees to pay his college tuition.He purchases trees for $10 each and sells them for $25 each. The number of trees he can sell is normally distributed with a mean of 100 and standard deviation of 30. How many trees should Joe purchase?
2 Condo Construction Company is going to First National Bank for a loan. At the present time, the bank is willing to lend Condo up to $1 million, with interest costs of 10%. Condo believes that the amount of borrowed funds needed during the current year is normally distributed, with a mean of
1 a In Example 3, why is it unrealistic to assume that the distribution of the number of no-shows is independent of q?b If the number of no-shows were normally distributed with a mean of .05q and a standard deviation of .05q, would we still have a news vendor problem?
The ticket price for a New York–Indianapolis flight is $200. Each plane can hold up to 100 passengers. Usually, some of the passengers who have purchased tickets for a flight fail to show up (no-shows). To protect against no-shows, the airline will try to sell more than 100 tickets for each
The American Bar Association (ABA) is holding its annual convention in Las Vegas. Six months before the convention begins, the ABA must decide how many rooms should be reserved in the convention hotel. At this time, the ABA can reserve rooms at a cost of $50 per room, but six months before the
9 Some universities allow an employee to put an amount q into an account at the beginning of each year, to be used for child-care expenses. The amount q is not subject to federal income tax. Assume that all other income is taxed by the federal government at a 40% rate. If child-care expenses for
4 If co is fixed, will an increase in cu increase or decrease the optimal order quantity?
3 If cu is fixed, will an increase in co increase or decrease the optimal order quantity?
In August, Walton Bookstore must decide how many of next year’s nature calendars should be ordered. Each calendar costs the bookstore $2 and is sold for $4.50. After January 1, any unsold calendars are returned to the publisher for a refund of 75¢ per calendar.Walton believes that the number of
1 Pizza King and Noble Greek are two competing restaurants. Each must determine simultaneously whether to undertake small, medium, or large advertising campaigns.Pizza King believes that it is equally likely that Noble Greek will undertake a small, a medium, or a large advertising campaign. Given
c Explain why marginal analysis fails to find the value of q that minimizes E(q). 1 Suppose E(q) is E(0) = 8, E(1) = 6, E(2) = 5, E(3) = 7, E(4) =6, E(5) = 5.5, E(6) = 4.5, and E(7) = 5.
b If marginal analysis is used to determine the value of q that minimizes E(q), what is the answer? 1 Suppose E(q) is E(0) = 8, E(1) = 6, E(2) = 5, E(3) = 7, E(4) =6, E(5) = 5.5, E(6) = 4.5, and E(7) = 5.
Patty is trying to determine which of two courses to take. If she takes the operations research course, she believes that she has a 10% chance of receiving an A, a 40% chance for a B, and a 50% chance for a C. If Patty takes a statistics course, she has a 70% chance for a B, a 25% chance for a C,
1 Suppose E(q) is E(0) = 8, E(1) = 6, E(2) = 5, E(3) = 7, E(4) =6, E(5) = 5.5, E(6) = 4.5, and E(7) = 5.a What value of q minimizes E(q)?
15 Although the Von Neumann–Morgenstern axioms seem plausible, there are many reasonable situations in which people appear to violate these axioms. For example, supposea recent college graduate must choose between three job offers on the basis of starting salary, location of job, and opportunity
5 A company produces three types of items. A single machine is used to produce the three items on a cyclical basis. The company has the policy that every item is produced once during each cycle, and it wants to determine the number of production cycles per year that will minimize the sum of holding
2 Suppose Joe’s is considering manufacturing desks. It costs $250 to set up a production run, and Joe’s has the capacity to manufacture up to 10,000 desks per year. What is the optimal production run size? How many production runs will be made each year?
1 Customers at Joe’s Office Supply Store demand an average of 6,000 desks per year. Each time an order is placed, an ordering cost of $300 is incurred. The annual holding cost for a single desk is 25% of the $200 cost of a desk. One week elapses between the placement of an order and the arrival
4 The Nitro Fertilizer Company is developing a new fertilizer. If Nitro markets the product and it is successful, the company will earn a $50,000 profit; if it is unsuccessful, the company will lose $35,000. In the past, similar products have been successful 60% of the time. At a cost of $5,000,
2 In Problem 1, suppose that Square City manufactures the washing machines. The company can manufacture washing machines at a rate of 30,000 per year. What manufacturing policy will minimize the annual cost of meeting demand?
1 Observed demand for air conditioners during the last four quarters was as follows: fall, 100; winter, 50; spring, 150; summer, 300. Is it reasonable to use an EOQ model in this situation?
4 For the model developed in this section, determine a the average length of time it takes to meet demand for a unit.b the fraction of all demanded units that are backordered.
3 Suppose that instead of measuring shortage in terms of cost per shortage year, a cost of S dollars is incurred for each unit the firm is short. This cost does not depend on the length of time before the backlogged demand is satisfied.Determine a new expression for TC(q, M), and explain how to
2 A Mercedes dealer must pay $20,000 for each car purchased. The annual holding cost is estimated to be 25%of the dollar value of inventory. The dealer sells an average of 500 cars per year. He believes that demand is backlogged but estimates that if he is short one car for one year he will lose
1 Show that the optimal order quantity for the backlogged demand model is always at least as large as the EOQ but that the maximum inventory level for the backlogged demand model cannot exceed the EOQ.
Each year, the Smalltown Optometry Clinic sells 10,000 frames for eyeglasses. The clinic orders frames from a regional supplier, which charges $15 per frame. Each order incurs an ordering cost of $50. Smalltown Optometry believes that the demand for frames can be backlogged and that the cost of
4 A company has the option of purchasing a good or manufacturing the item. If the item is purchased, the company will be charged $25 per unit plus a cost of $4 per order. If the company manufactures the item, it has a production capacity of 8,000 units per year. It costs $50 to set up a production
3 The production process at Father Dominic’s Pizza can produce 400 pizza pies per day; the firm operates 250 days per year. Father Dominic’s has a cost of $180 per production run and a holding cost of $5 per pizza-year. The pies are frozen immediately after they are produced and stored in a
2 A company can produce 100 home computers per day.The setup cost for a production run is $1,000. The cost of holding a computer in inventory for one year is $300.Customers demand 2,000 home computers per month(assume that 1 month 30 days and 360 days 1 year).What is the optimal production run
1 Show that the optimal run size always exceeds the EOQ.Give an intuitive explanation for this result.
Macho Auto Company needs to produce 10,000 car chassis per year. Each is valued at$2,000. The plant has the capacity to produce 25,000 chassis per year. It costs $200 to set up a production run, and the annual holding cost is 25¢, per dollar of inventory. Determine the optimal production run size.
5 In Problem 4, suppose the cost per order is $1 and the monthly demand is 50 thermometers. What is the optimal order quantity? How small a discount could the supplier offer and still have the hospital accept the discount?
Each time an order is placed for disks, how many boxes of disks should be ordered? How many orders will be placed annually? What is the total annual cost of meeting the accounting firm’s disk needs?
16 The borough of Staten Island has two “sanitation districts.” In district 1, street litter piles up at an average rate of 2,000 tons per week, and in district 2 at an average rate of 1,000 tons per week. Each district has 500 miles of streets. Staten Island has 10 sanitation crews and each
15 Show that the reorder point in the EOQ model equals the remainder when LD is divided by the EOQ.
14 Consider a firm that knows that the price of the product it is ordering is going to increase permanently by $X. How much of the product should be ordered before the price increase goes into effect?Here is one approach to this question: Suppose the firm orders Q units before the price increase
13 A newspaper has 500,000 subscribers who pay $4 per month for the paper. It costs the company $200,000 to bill all its customers. Assume that the company can earn interest at a rate of 20% per year on all revenues. Determine how often the newspaper should bill its customers. (Hint: Look at unpaid
12 During each year, CSL Computer Company needs to train 27 service representatives. No matter how many students are trained, it costs $12,000 to run a training program. Since service reps earn a monthly salary of $1,500,CSL does not want to train them before they are needed.Each training session
11 A drug store sells 30 bottles of antibiotics per week.Each time it orders antibiotics, there is a fixed ordering cost of $10 and a cost of $10/bottle. Assume that the annual holding cost is 20% of the cost of a bottle of antibiotics, and suppose antibiotics spoil and cannot be sold if they spend
10 In terms of K, D, and h, what is the average length of time that an item spends in inventory before being used to meet demand? Explain how this result can be used to characterize a fast- or slow-moving item.
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