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operations research an introduction
Operations Research Applications And Algorithms 4th Edition Wayne L. Winston - Solutions
5 Ethan Sherwood owns two printing presses, on which he prints two types of jobs. At the beginning of each day, there is a.5 probability that a type 1 job will arrive, a .1 probability that a type 2 job will arrive, and a .4 probability that no job will arrive. Ethan receives $400 for completing a
4 During any day, I may own either 0 or 1 share of a stock.The price of the stock is governed by the Markov chain shown in Table 11. At the beginning of a day in which I own a share of stock, I may either sell it at today’s price or keep it. At the beginning of a day in which I don’t own a
3 Suppose you are using the policy iteration method to determine an optimal policy for an MDP. How might you use LINDO to solve the value determination equations?
2 Priceler Auto Corporation must determine whether or not to give consumers 8% or 11% financing on new cars. If Priceler gives 8% financing during the current month, the probability distribution of sales during the current month will be as shown in Table 9. If Priceler gives 11% financing during
1 A warehouse has an end-of-period capacity of 3 units.During a period in which production takes place, a setup cost of $4 is incurred. A $1 holding cost is assessed against each unit of a period’s ending inventory. Also, a variable production cost of $1 per unit is incurred. During each period,
At the beginning of each week, a machine is in one of four conditions (states): excellent(E), good (G), average (A), or bad (B). The weekly revenue earned by a machine in each type of condition is as follows: excellent, $100; good, $80; average, $50; bad, $10. After observing the condition of a
15 In Example 9, assume T 10 and pt |t|/10. Using a spreadsheet, solve for Robert’s optimal strategy.
14 In Example 10, suppose pt t/60 and dt t. Using a spreadsheet, solve for Dirk’s optimal strategy. (Hint: The possible states are 50, 51, . . . , 1,880 (thousands).)
13 At 7 A.M., eight people leave their cars for repair at Harry’s Auto Repair Shop. If person i’s car is ready by time t (7 A.M. time 0, and so on), he will pay Harry ri(t) dollars.For example, if person 2’s car must be ready by 2 P.M., we may have r2(8) 0. Harry estimates that with
12 Bailey Bliss is the campaign manager for Walter Glenn’s presidential campaign. He has D dollars to allocate to T winner-take-all primaries. If xt dollars are allocated to primary t, then with probability pt(xt), Glenn will win primary t and obtain vt delegates. With probability 1 pt(xt),
11 Georgia Stein is the new owner of the New York Yankees. Each season, Georgia must decide how much money to spend on the free agent draft. During each season, Georgia can spend any amount of money on free agents up to the team’s capital position at the beginning of the season.If the Yankees
10 An advertising firm has D dollars to spend on reaching customers in T separate markets. Market t consists of kt people. If x dollars are spent on advertising in market t, the probability that a given person in market t will be reached is pt(x). Each person in market t who is reached will buy ct
9 Martha and Ken Allen want to sell their house. At the beginning of each day, they receive an offer. We assume that from day to day, the sizes of the offers are independent random variables and that the probability that a given day’s offer is for j dollars is pj . An offer may be accepted during
8 Consider the following probabilistic inventory model:a At the beginning of each period, a firm observes its inventory position.b Then the firm decides how many units to produce during the current period. It costs c(x) dollars to produce x units during a period.c With probability q(d), d units are
7 The National Cat Foundling Home encourages people to adopt its cats, but (because of limited funds) it allows each prospective owner to inspect only four cats before choosing one of them to take home. Ten-year-old Sara is eager to adopt a cat and agrees to abide by the following rules. A randomly
6 I own a single share of Wivco stock. I must sell my share at the beginning of one of the next 30 days. Each day, the price of the stock changes. With probability q(x), the price tomorrow will increase by x% over today’s stock price(x can be negative). For example, with probability q(5),
5 At the beginning of each week, a machine is either running or broken down. If the machine runs throughout the week, it earns revenues of $100. If the machine breaks down during a week, it earns no revenue for that week. If the machine is running at the beginning of the week, we may perform
4 In the time interval between t and t 1 seconds before the departure of Braneast Airlines Flight 313, there is a probability pt that the airline will receive a reservation for the flight and a probability 1 pt that the airline will receive no reservation. The flight can seat up to 100
3 Consider a machine that may be in any one of the states 0, 1, 2, . . . . At the beginning of each month, the state of the machine is observed, and it is decided whether to replace or keep the machine. If the machine is replaced, a new state 0 machine arrives instantaneously. It costs R dollars to
2 At the beginning of each year, a firm observes its asset position (call itd) and may invest any amount x (0 x d) in a risky investment. During each year, the money invested doubles with probability p and is completely lost with probability 1 p. Independently of this investment, the firm’s
1 The space shuttle is about to go up on another flight.With probability pt(z), it will use z type t fuel cells during the flight. The shuttle has room for at most W fuel cells. If at any time during the flight, all the type t fuel cells burn out, a cost ct will be incurred. Assuming the goal is to
During month t(t 1, 2, . . . , 60), expert safecracker Dirk Stack knows that he will be offered a role in a bank job that will pay him dt dollars. There is, however, a probability pt that month t’s job will result in his capture. If Dirk is captured, all his money will be lost. Dirk’s goal is
E. J. Korvair Department Store is trying to determine an optimal cash management policy.During each day, the demand for cash may be described by a random variable D, where p(D d) p(d). At the beginning of each day, the store sends an employee to the bank to deposit or withdraw funds. Each bank
When Sally Mutton arrives at the bank, 30 minutes remain on her lunch break. If Sally makes it to the head of the line and enters service before the end of her lunch break, she earns reward r. However, Sally does not enjoy waiting in lines, so to reflect her dislike for waiting in line, she incurs
Each year, the owner of a lake must determine how many bass to capture and sell. During year t, a price pt will be received for each bass that is caught. If the lake contains b bass at the beginning of year t, the cost of capturing x bass is ct (x|b). Between the time that year t’s bass are
Sunco Oil has D dollars to allocate for drilling at sites 1, 2, . . . , T. If x dollars are allocated to site t, the probability is qt (x) that oil will be found on site t. Sunco estimates that if site t has any oil, it is worth rt dollars. Formulate a recursion that could be used to enable Sunco
3 Supppose that Army trails by 14 points in the Army–Navy football game. Army’s guardian angel has assured the Army coach that his team will have the ball two more times during the game and will score a touchdown (worth 6 points) each time it has the ball. The Army coach has also been assured
1 Vladimir Ulanowsky is playing Keith Smithson in a two-game chess match. Winning a game scores 1 match match is over 1/2 . Explain this anomalous result.2 Dickie Hustler has $2 and is going to toss an unfair coin (probability .4 of heads) three times. Before each toss, he can bet any amount of
Martina McEnroe has two types of serves: a hard serve (H) and a soft serve (S).† The probability that Martina’s hard serve will land in bounds is pH, and the probability that her soft serve will land in bounds is pS. If Martina’s hard serve lands in bounds, there is a probability wH that
A gambler has $2. She is allowed to play a game of chance four times, and her goal is to maximize her probability of ending up with a least $6. If the gambler bets b dollars on a play of the game, then with probability .40, she wins the game and increases her capital position by b dollars; with
4 Chip Bilton sells sweatshirts at State U football games.He is equally likely to sell 200 or 400 sweatshirts at each game. Each time Chip places an order, he pays $500 plus$5 for each sweatshirt he orders. Each sweatshirt sells for$8. A holding cost of $2 per shirt (because of the opportunity cost
3 In Example 2, suppose that shortages are allowed, and each shortage results in a lost sale and a cost incurred of $3.Now re-solve Example 2.
2 Re-solve Example 2 if the end-of-period holding cost is$2 per unit.
1 For Example 2, suppose that the period 1 demand is 1 unit, and the period 2 demand is 2 units. What would be the optimal production schedule?
Consider the following three-period inventory problem. At the beginning of each period, a firm must determine how many units should be produced during the current period. During a period in which x units are produced, a production cost c(x) is incurred, where c(0) 0, and for x 0, c(x) 3 2x.
2 Suppose that $4 million is available for investment in three projects. The probability distribution of the net present value earned from each project depends on how much is invested in each project. Let It be the random variable denoting the net present value earned by project t. The distribution
1 In Example 1, find another allocation of milk that maximizes expected daily revenue.
11 At the beginning of month t, farmer Smith has xt bushels of wheat in his warehouse. He has the opportunity to sell wheat at a price st dollars per bushel and can buy wheat at pt dollars per bushel. Farmer Smith’s warehouse can hold at most C units at the end of each month.a Formulate a
8 At the beginning of each year, Barnes Carr Oil sets the world oil price. If a price p is set, then D(p) barrels of oil will be demanded by world customers. We assume that during any year, each oil company sells the same number of barrels of oil. It costs Barnes Carr Oil c dollars to extract and
7 A company needs the following number of workers during each of the next five years: year 1, 15; year 2, 30;year 3, 10; year 4, 30; year 5, 20. At present, the company has 20 workers. Each worker is paid $30,000 per year. At the beginning of each year, workers may be hired or fired.It costs
5 I am working as a cashier at the local convenience store.A customer’s bill is $1.09, and he gives me $2.00. I want to give him change using the smallest possible number of coins. Use dynamic programming to determine how to give the customer his change. Does the answer suggest a general result
3 Reconsider Problem 2, but now suppose that demands need not be met on time. Assume that all lost demand is backlogged and that a $1 per-unit shortage cost is assessed against the number of shortages incurred during each month.All demand must be met by the end of month 4. Use dynamic programming
2 A company must meet the following demands on time:month 1, 1 unit; month 2, 1 unit; month 3, 2 units; month 4, 2 units. It costs $4 to place an order, and a $2 per-unit holding cost is assessed against each month’s ending inventory. At the beginning of month 1, 1 unit is available.Orders are
1 In the network in Figure 14, find the shortest path from node 1 to node 10 and the shortest path from node 2 to node 10.
3 Use a spreadsheet to solve Problem 5 of Section 18.4.
2 Use a spreadsheet to solve Problem 4 of Section 18.4.
1 Use a spreadsheet to solve Problem 2 of Section 18.3.
4 Explain why HC(t)/t tends to be an increasing function of t.
3 Use the Wagner–Whitin and Silver–Meal methods to find production schedules for the following dynamic lotsize problem: K $30, h $1, d1 40, d2 60, d3 10, d4 70, d5 20.
2 Use the Wagner–Whitin and Silver–Meal methods to find production schedules for the following dynamic lotsize problem: K $50, h $0.40, d1 10, d2 60, d3 20, d4 140, d5 90.
1 For Example 15, suppose we had an inventory of 200 units. What would be the optimal production schedule?What if the initial inventory were 400 units?
We now determine an optimal production schedule for a five-period dynamic lot-size model with K $250, c $2, h $1, d1 220, d2 280, d3 360, d4 140, and d5 270. We assume that the initial inventory level is zero. The solution to this example is given later in this section.
11 The state of Transylvania consists of three cities with the following populations: city 1, 1.2 million people; city 2, 1.4 million people; city 3, 400,000 people. The Transylvania House of Representatives consists of three representatives.Given proportional representation, city 1 should have d1
10 During month t, a firm faces a demand for dt units of a product. The firm’s production cost during month t consists of two components. First, for each unit produced during month t, the firm incurs a variable production cost of ct.Second, if the firm’s production level during month t 1 is
9 Public Service Indiana (PSI) is considering five possible locations to build power plants during the next 20 years. It will cost ci dollars to build a plant at site i and hi dollars to operate a site i plant for a year. A plant at site i can supply ki kilowatthours(kwh) of generating capacity.
8 MacBurger has just opened a fast-food restaurant in Bloomington. Currently, i0 customers frequent MacBurger(we call these loyal customers), and Ni0 customers frequent other fast-food establishments (we call these nonloyal customers). At the beginning of each month, MacBurger must decide how much
7 Farmer Jones is trying to raise a prize steer for the Bloomington 4-H show. The steer now weighs w0 pounds.Each week, farmer Jones must determine how much food to feed the steer. If the steer weighs w pounds at the beginning of a week and is fed p pounds of food during a week, then at the
6 E.T. is about to fly home. For the trip to be successful, the ship’s solar relay, warp drive, and candy maker must all function properly. E.T. has found three unemployed actors who are willing to help get the ship ready for takeoff. Table 18 gives, as a function of the number of actors assigned
4 Develop a dynamic programming formulation of the CSL Computer problem of Section 3.12.
3 Assume that during minute t (the current minute is minute 1), the following sequence of events occurs: (1) At the beginning of the minute, xt customers arrive at the cash register; (2) the store manager decides how many cash registers should be operated during the current minute;(3) if s cash
2 At the beginning of year 1, Julie Ripe has D dollars (this includes year 1 income). During each year, Julie earns i dollars and must determine how much money she should consume and how much she should invest in Treasury bills. During a year in which Julie consumes d dollars, she earns a utility
1 At the beginning of year 1, Sunco Oil owns i0 barrels of oil reserves. During year t(t 1, 2, . . . , 10), the following events occur in the order listed: (1) Sunco extracts and refines x barrels of oil reserves and incurs a cost c(x): (2)Sunco sells year t’s extracted and refined oil at a
Glueco is planning to introduce a new product in three different regions. Current estimates are that the product will sell well in each region with respective probabilities .6, .5, and.3. The firm has available two top sales representatives that it can send to any of the three regions. The
Joe Cougar needs to drive from city 1 to city 10. He is no longer interested in minimizing the length of his trip, but he is interested in minimizing the maximum altitude above sea level that he will encounter during his drive. To get from city 1 to city 10, he must follow a path in Figure 10. The
The traveling salesperson problem (see Section 9.6) can be solved by using dynamic programming.As an example, we solve the following traveling salesperson problem: It’s the last weekend of the 2004 election campaign, and candidate Walter Glenn is in New York City. Before election day, Walter must
Sunco Oil needs to build enough refinery capacity to refine 5,000 barrels of oil per day and 10,000 barrels of gasoline per day. Sunco can build refinery capacity at four locations.The cost of building a refinery at site t that has the capacity to refine x barrels of oil per day and y barrels of
Farmer Jones now possesses $5,000 in cash and 1,000 bushels of wheat. During month t, the price of wheat is pt. During each month, he must decide how many bushels of wheat to buy (or sell). There are three restrictions on each month’s wheat transactions: (1) During any month, the amount of money
An electric power utility forecasts that rt kilowatt-hours (kwh) of generating capacity will be needed during year t (the current year is year 1). Each year, the utility must decide by how much generating capacity should be expanded. It costs ct(x) dollars to increase generating capacity by x kwh
The owner of a lake must decide how many bass to catch and sell each year. If she sells x bass during year t, then a revenue r (x) is earned. The cost of catching x bass during a year is a function c(x,b) of the number of bass caught during the year and ofb, the number of bass in the lake at the
1 Use Equations (10)–(10.3) to determine an optimal replacement policy for the engine analyzer example.
3 The knapsack problem of Problem 2 can be viewed as finding the longest route in a particular network.a Draw the network corresponding to the recursion derived from (7).b Draw the network corresponding to the recursion derived from (8).
Suppose a 10-lb knapsack is to be filled with the items listed in Table 9. To maximize total benefit, how should the knapsack be filled?
3 In Figure 5, determine the production level and cost associated with each of the following arcs:a (2, 3)–(3, 1)b (4, 2)–(5, 0)
2 An electronics firm has a contract to deliver the following number of radios during the next three months;month 1, 200 radios; month 2, 300 radios; month 3, 300 radios. For each radio produced during months 1 and 2, a$10 variable cost is incurred; for each radio produced during month 3, a $12
1 In Example 4, determine the optimal production schedule if the initial inventory is 3 units.
A company knows that the demand for its product during each of the next four months will be as follows: month 1, 1 unit; month 2, 3 units; month 3, 2 units; month 4, 4 units.At the beginning of each month, the company must determine how many units should be produced during the current month. During
Joe Cougar lives in New York City, but he plans to drive to Los Angeles to seek fame and fortune. Joe’s funds are limited, so he has decided to spend each night on his trip at a friend’s house. Joe has friends in Columbus, Nashville, Louisville, Kansas City, Omaha, Dallas, San Antonio, and
4 Given a 7-oz cup and a 3-oz cup, explain how we can return from a well with 5 oz of water.
3 We have 21 coins and are told that one is heavier than any of the other coins. How many weighings on a balance will it take to find the heaviest coin? (Hint: If the heaviest coin is in a group of three coins, we can find it in one weighing. Then work backward to two weighings, and so on.)
2 Three players have played three rounds of a gambling game. Each round has one loser and two winners. The losing player must pay each winner the amount of money that the winning player had at the beginning of the round. At the end of the three rounds each player has $10. You are told that each
1 Suppose there are 40 matches on a table. I begin by picking up 1, 2, 3, or 4 matches. Then my opponent must pick up 1, 2, 3, or 4 matches. We continue until the last match is picked up. The player who picks up the last match is the loser. Can I be sure of victory? If so, how?
Suppose there are 30 matches on a table. I begin by picking up 1, 2, or 3 matches. Then my opponent must pick up 1, 2, or 3 matches. We continue in this fashion until the last match is picked up. The player who picks up the last match is the loser. How can I (the first player) be sure of winning
15 General Motors has three auto divisions (1, 2, and 3). It also has an accounting division and a management consulting division. The question is: What fraction of the cost of the accounting and management consulting divisions should be allocated to each auto division? We assume that the entire
14 Consider an LP with five basic feasible solutions and a unique optimal solution. Assume that the simplex method begins at the worst basic feasible solution, and on each pivot the simplex is equally likely to move to any better basic feasible solution.On the average, how many pivots will be
12 For a Q matrix representing the transitions between transient states in an absorbing Markov chain, it can be shown that(I Q)1 I Q Q2 Qn a Explain why this expression for (I Q)1 is plausible.b Define mij expected number of periods spent in transient state tj before
11 Freezco, Inc., sells refrigerators. The company has issued a warranty on all refrigerators that requires free replacement of any refrigerator that fails before it is three years old. We are given the following information: (1) 3% of all new refrigerators fail during their first year of
9 In the gambler’s ruin problem (Example 1), assume p .60.a What is the probability that I reach $4?b What is the probability that I am wiped out?c What is the expected duration of the game?
8 Suppose each box of Corn Snaps cereal contains one of five different Harry Potter trading cards. On the average, how many boxes of cereal will I have to buy to obtain a complete set of trading cards?
7 Suppose I toss a fair coin, and the first toss comes up heads. If I keep tossing the coin until I either see two consecutive heads or two consecutive tails, what is the probability that I will see two consecutive heads before I see two consecutive tails?
6 I now have $2, and my goal is to have $6. I will repeatedly flip a coin that has a .4 chance of coming up heads. If the coin comes up heads, I win the amount I bet. If the coin comes up tails, I lose the amount of my bet. Let us suppose I follow the bold strategy of betting Min($6 current asset
4 Absorbing Markov chains are used in marketing to model the probability that a customer who is contacted by telephone will eventually buy a product. Consider a prospective customer who has never been called about purchasing a product. After one call, there is a 60% chance that the customer will
3 A forest consists of two types of trees: those that are 0–5 ft and those that are taller than 5 ft. Each year, 40% of all 0–5-ft tall trees die, 10% are sold for $20 each, 30% stay between 0 and 5 ft, and 20% grow to be more than 5 ft. Each year, 50%of all trees taller than 5 ft are sold for
2 The Herald Tribble has obtained the following information about its subscribers: During the first year as subscribers, 20%of all subscribers cancel their subscriptions. Of those who have subscribed for one year, 10% cancel during the second year.Of those who have been subscribing for more than
1 What is the average length of time that a newly hired junior lawyer spends working for the firm?2 What is the probability that a junior lawyer makes it to partner?3 What is the average length of time that a partner spends with the firm (as a partner)?
1 What is the probability that a new account will eventually be collected?2 What is the probability that a one-month-overdue account will eventually become a bad debt?3 If the firm’s sales average $100,000 per month, how much money per year will go uncollected?
The law firm of Mason and Burger employs three types of lawyers: junior lawyers, senior lawyers, and partners. During a given year, there is a .15 probability that a junior lawyer will be promoted to senior lawyer and a .05 probability that he or she will leave the firm.Also, there is a .20
The accounts receivable situation of a firm is often modeled as an absorbing Markov chain.† Suppose a firm assumes that an account is uncollectable if the account is more than three months overdue. Then at the beginning of each month, each account may be classified into one of the following
15 The Gotham City Maternity Ward contains 2 beds. Admissions are made only at the beginning of the day. Each day, there is a .5 probability that a potential admission will arrive. A patient can be admitted only if there is an open bed at the beginning of the day. Half of all patients are
14 Each month, customers are equally likely to demand 1 or 2 computers from a Pearco dealer. All orders must be met from current stock. Two ordering policies are under consideration:Policy 1 If ending inventory is 2 units or less, order enough to bring next month’s beginning inventory to 4
13 An important machine is known to never last more than four months. During its first month of operation, it fails 10% of the time. If the machine completes its first month, then it fails during its second month 20% of the time. If the machine completes its second month of operation, then it will
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