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Introduction to Operations Research 10th edition Frederick S. Hillier, Gerald J. Lieberman - Solutions
Consider a Jackson network with three service facilities having the parameter values shown below.T (a) Find the total arrival rate at each of the facilities.
When describing economic analysis of the number of servers to provide in a queueing system, Sec. 17.10 introduces a basic cost model where the objective is to minimize E(TC) = Css + CwL. The purpose of this problem is to enable you to explore the effect that the relative sizes of Cs and Cw have on
Show thatby using the statistical definitions of L and Lq in terms of the Pn.
Jim McDonald, manager of the fast-food hamburger restaurant McBurger, realizes that providing fast service is a key to the success of the restaurant. Customers who have to wait very long are likely to go to one of the other fast-food restaurants in town next time. He estimates that each minute a
The Garrett-Tompkins Company provides three copy machines in its copying room for the use of its employees. However, due to recent complaints about considerable time being wasted waiting for a copier to become free management is considering adding one or more additional copy machines. During the
Jim Wells, vice-president for manufacturing of the Northern Airplane Company, is exasperated. His walk through the company’s most important plant this morning has left him in a foul mood. However, he now can vent his temper at Jerry Carstairs, the plant’s production manager, who has just been
Many angry customers are complaining about the long waits needed to get through to a call center. It appears that more service representatives are needed to answer the calls. Another option is to train the service representatives further to enable them to answer calls more efficiently. Some
Consider a typical barber shop. Demonstrate that it is a queueing system by describing its components.
Suppose that a queueing system has two servers, an exponential interarrival time distribution with a mean of 2 hours, and an exponential service-time distribution with a mean of 2 hours for each server. Furthermore, a customer has just arrived at 12:00 noon. (a) What is the probability that the
The jobs to be performed on a particular machine arrive according to a Poisson input process with a mean rate of two per hour. Suppose that the machine breaks down and will require 1 hour to be repaired. What is the probability that the number of new jobs that will arrive during this time is (a) 0,
The time required by a mechanic to repair a machine has an exponential distribution with a mean of 4 hours. However, a special tool would reduce this mean to 2 hours. If the mechanic repairs a machine in less than 2 hours, he is paid $100; otherwise, he is paid $80. Determine the mechanic’s
A three-server queueing system has a controlled arrival process that provides customers in time to keep the servers continuously busy. Service times have an exponential distribution with mean 0.5. You observe the queueing system starting up with all three servers beginning service at time t = 0.
A queueing system has three servers with expected service times of 20 minutes, 15 minutes, and 10 minutes. The service times have an exponential distribution. Each server has been busy with a current customer for 5 minutes. Determine the expected remaining time until the next service completion.
Consider a queueing system with two types of customers. Type 1 customers arrive according to a Poisson process with a mean rate of 5 per hour. Type 2 customers also arrive according to a Poisson process with a mean rate of 5 per hour. The system has two servers, both of which serve both types of
Consider a two-server queueing system where all service times are independent and identically distributed according to an exponential distribution with a mean of 10 minutes. Service is provided on a first-come-first-served basis. When a particular customer arrives, he finds that both servers are
For each of the following statements regarding service times modeled by the exponential distribution, label the statement as true or false and then justify your answer by referring to specific statements in the chapter. (a) The expected value and variance of the service times are always equal. (b)
As for Property 3 of the exponential distribution, let T1, T2, . . . , Tn be independent exponential random variables with parameters α1, α2, . . . , αn, respectively, and let U = min{T1, T2, . . . , Tn}. Show that the probability that a particular random
Consider the birth-and-death process with all μn = 2 (n = 1, 2, . . .), λ0 = 3, λ1 = 2, λ2 = 1, and λn = 0 for n = 3, 4, . . . . (a) Display the rate diagram. (b) Calculate P0, P1, P2, P3, and Pn for n = 4, 5, . . . . (c) Calculate L, Lq, W, and Wq.
Newell and Jeff are the two barbers in a barber shop they own and operate. They provide two chairs for customers who are waiting to begin a haircut, so the number of customers in the shop varies between 0 and 4. For n = 0, 1, 2, 3, 4, the probability Pn that exactly n customers are in the shop is
Consider a birth-and-death process with just three attainable states (0, 1, and 2), for which the steady-state probabilities are P0, P1, and P2, respectively. The birth-and-death rates are summarized in the following table:(a) Construct the rate diagram for this birth-and-death process. (b) Develop
Consider the birth-and-death process with the following mean rates. The birth rates are λ0 = 2, λ1 = 3, λ2 = 2, λ3 = 1, and λn = 0 for n > 3. The death rates are μ1 = 3, μ2 = 4, μ3 = 1, and μn = 2 for n > 4. (a) Construct the rate diagram for this birth-and-death process. (b) Develop the
Consider the birth-and-death process with all λn = 2 (n = 0, 1, . . .), μ1 = 2, and μn = 4 for n = 2, 3, . . . . (a) Display the rate diagram. (b) Calculate P0 and P1. Then give a general expression for Pn in terms of P0 for n = 2, 3, . . .. (c) Consider a queueing system with two servers that
A service station has one gasoline pump. Cars wanting gasoline arrive according to a Poisson process at a mean rate of 15 per hour. However, if the pump already is being used, these potential customers may balk (drive on to another service station). In particular, if there are n cars already at the
A maintenance person has the job of keeping two machines in working order. The amount of time that a machine works before breaking down has an exponential distribution with a mean of 10 hours. The time then spent by the maintenance person to repair the machine has an exponential distribution with a
Consider a single-server queueing system where interarrival times have an exponential distribution with parameter λ and service times have an exponential distribution with parameter μ. In addition, customers renege (leave the queueing system without being served) if their waiting time in the
A certain small grocery store has a single checkout stand with a full-time cashier. Customers arrive at the stand “randomly” (i.e., a Poisson input process) at a mean rate of 30 per hour. When there is only one customer at the stand, she is processed by the cashier alone, with an expected
A department has one word-processing operator. Documents produced in the department are delivered for word processing according to a Poisson process with an expected interarrival time of 20 minutes. When the operator has just one document to process, the expected processing time is 15 minutes. When
Customers arrive at a queueing system according to a Poisson process with a mean arrival rate of 2 customers per minute. The service time has an exponential distribution with a mean of 1 minute. An unlimited number of servers are available as needed so customers never wait for service to begin.
Suppose that a single-server queueing system fits all the assumptions of the birth-and-death process except that customers always arrive in pairs. The mean arrival rate is 2 pairs per hour (4 customers per hour) and the mean service rate (when the server is busy) is 5 customers per hour. (a)
Mom-and-Pop’s Grocery Store has a small adjacent parking lot with three parking spaces reserved for the store’s customers. During store hours, cars enter the lot and use one of the spaces at a mean rate of 2 per hour. For n = 0, 1, 2, 3, the probability Pn that exactly n spaces currently are
Consider a single-server queueing system with a finite queue that can hold a maximum of 2 customers excluding any being served. The server can provide batch service to 2 customers simultaneously, where the service time has an exponential distribution with a mean of 1 unit of time regardless of the
Consider a queueing system that has two classes of customers, two clerks providing service, and no queue. Potential customers from each class arrive according to a Poisson process, with a mean arrival rate of 10 customers per hour for class 1 and 5 customers per hour for class 2, but these arrivals
The Dolomite Corporation is making plans for a new factory. One department has been allocated 12 semiautomatic machines. A small number (yet to be determined) of operators will be hired to provide the machines the needed occasional servicing (loading, unloading, adjusting, setup, and so on). A
A shop contains three identical machines that are subject to a failure of a certain kind. Therefore, a maintenance system is provided to perform the maintenance operation (recharging) required by a failed machine. The time required by each operation has an exponential distribution with a mean of 30
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 17.6. Briefly describe how queueing theory was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study
The 4M Company has a single turret lathe as a key work center on its factory floor. Jobs arrive at this work center according to a Poisson process at a mean rate of 2 per day. The processing time to perform each job has an exponential distribution with a mean of 1/4 day. Because the jobs are bulky,
Customers arrive at a single-server queueing system according to a Poisson process at a mean rate of 10 per hour. If the server works continuously, the number of customers that can be served in an hour has a Poisson distribution with a mean of 15. Determine the proportion of time during which no
Consider the M/M/1 model, with λ < μ. (a) Determine the steady-state probability that a customer’s actual waiting time in the system is longer than the expected waiting time in the system, i.e., P {W > W}. (b) Determine the steady-state probability that a customer’s actual waiting time in the
Suppose that the demand for a product is 30 units per month and the items are withdrawn at a constant rate. The setup cost each time a production run is undertaken to replenish inventory is $15. The production cost is $1 per item, and the inventory holding cost is $0.30 per item per month. (a)
You have been hired as an operations research consultant by a company to reevaluate the inventory policy for one of its products. The company currently uses the basic EOQ model. Under this model, the optimal order quantity for this product is 1,000 units, so the maximum inventory level also is
In the basic EOQ model, suppose the stock is replenished uniformly (rather than instantaneously) at the rate of b items per unit time until the order quantity Q is fulfilled. Withdrawals from the inventory are made at the rate of a items per unit time, where a(a) Find the total cost per unit time
MBI is a manufacturer of personal computers. All its personal computers use a hard disk drive which it purchases from Ynos. MBI operates its factory 52 weeks per year, which requires assembling 100 of these disk drives into computers per week. MBI€™s annual holding cost rate is 20 percent
The Gilbreth family drinks a case of Royal Cola every day, 365 days a year. Fortunately, a local distributor offers quantity discounts for large orders as shown in the table below, where the price for each category applies to every case purchased. Considering the cost of gasoline, Mr. Gilbreth
Kenichi Kaneko is the manager of a production department which uses 400 boxes of rivets per year. To hold down his inventory level, Kenichi has been ordering only 50 boxes each time. However, the supplier of rivets now is offering a discount for higher-quantity orders according to the following
Sarah operates a concession stand at a downtown location throughout the year. One of her most popular items is circus peanuts, selling about 200 bags per month.Sarah purchases the circus peanuts from Peter€™s Peanut Shop. She has been purchasing 100 bags at a time. However, to encourage
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 18.5. Briefly describe how inventory theory was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Consider an inventory system that fits the model for a serial two-echelon system presented in Sec. 18.5, where K1 = $15,000, K2 = $500, h1 = $20, h2 = $22, and d = 5,000. Develop a table like Table 18.1 that shows the results from performing both separate optimization of the installations and
A company soon will begin production of a new product. When this happens, an inventory system that fits the model for a serial two-echelon system presented in Sec. 18.5 will be used. At this time, there is great uncertainty about what the setup costs and holding costs will be at the two
A company owns both a factory to produce its products and a retail outlet to sell them. A certain new product will be sold exclusively through this retail outlet. Its inventory of this product will be replenished when needed from the factory’s inventory, where an administrative and shipping cost
The demand for a product is 600 units per week, and the items are withdrawn at a constant rate. The setup cost for placing an order to replenish inventory is $25. The unit cost of each item is $3, and the inventory holding cost is $0.05 per item per week. (a) Assuming shortages are not allowed,
A company produces a certain product by assembling it at an assembly plant. All the components needed to assemble the product are purchased from a single supplier. A shipment of all the components is received from the supplier each time the assembly plant needs to replenish its inventory of the
Consider a three-echelon inventory system that fits the model for a serial multiechelon system presented in Sec. 18.5, where the model parameters for this particular system are given below.
Follow the instructions of Prob. 18.5-6 for a five-echelon inventory model fitting the corresponding model in Sec. 18.5, where the model parameters are given below.
Reconsider the example of a four-echelon inventory system presented in Sec. 18.5, where its model parameters are given in Table 18.2. Suppose now that the setup costs at the four installations have changed from what is given in Table 18.2, where the new values are K1 = $1,000, K2 = $5, K3 = $75,
One of the many products produced by the Global Corporation is marketed primarily in the United States. A rough form of the product is produced in one of the corporation’s plants in Asia and then is shipped to a plant in the United States for the finish work. The finished product next is sent to
Suppose that production planning is to be done for the next 5 months, where the respective demands are r1 = 2, r2 = 4, r3 = 2, r4 = 2, and r5 = 3. The setup cost is $4,000, the unit production cos is $1,000, and the unit holding cost is $300. Use the deterministic periodic-review model to determine
Reconsider the example used to illustrate the deterministic periodic-review model in Sec. 18.4. Solve this problem when the demands are increased by 1 airplane in each period.
Reconsider the example used to illustrate the deterministic periodic-review model in Sec. 18.4. Suppose that the following single change is made in the example. The cost of producing each airplane now varies from period to period. In particular, in addition to the setup cost of $2 million, the cost
Consider a situation where a particular product is produced and placed in in-process inventory until it is needed in a subsequent production process. The number of units required in each of the next 3 months, the setup cost, and the regular-time unit production cost (in units of thousands of
Henry Edsel is the owner of Honest Henry’s, the largest car dealership in its part of the country. His most popular car model is the Triton, so his largest costs are those associated with ordering these cars from the factory and maintaining an inventory of Tritons on the lot. Therefore, Henry has
Tim Madsen is the purchasing agent for Computer Center, a large discount computer store. He has recently added the hottest new computer, the Power model, to the store’s stock of goods. Sales of this model now are running at about 13 per week. Tim purchases these computers directly from the
One of the largest selling items in J.C. Ward’s Department Store is a new model of refrigerator that is highly energy-efficient. About 40 of these refrigerators are being sold per month. It takes about a week for the store to obtain more refrigerators from a wholesaler. The demand during this
When using the stochastic continuous-review model presented in Sec. 18.6, a difficult managerial judgment decision needs to be made on the level of service to provide to customers. The purpose of this problem is to enable you to explore the trade-off involved in making this decision. Assume that
The preceding problem describes the factors involved in making a managerial decision on the service level L to use. It also points out that for any given values of L, h (the unit holding cost per year), and σ (the standard deviation when the demand during the lead time has a normal distribution),
What is the effect on the amount of safety stock provided by the stochastic continuous-review model presented in Sec. 18.6 when the following change is made in the inventory system? (Consider each change independently.) (a) The lead time is reduced to 0 (instantaneous delivery). (b) The service
Jed Walker is the manager of Have a Cow, a hamburger restaurant in the downtown area. Jed has been purchasing all the restaurant’s beef from Ground Chuck (a local supplier) but is considering switching to Chuck Wagon (a national warehouse) because its prices are lower. Weekly demand for beef
Reconsider the Blue Skies Airlines example presented in Sec. 18.8. Regarding the flight under consideration, recent experience indicates that the demand for the very low discount fare of $200 is so high that it may be possible to considerably increase this fare and still usually fill up the
The most popular cruise offered by Luxury Cruises is a three-week cruise in the Mediterranean each July with daily ports of call at interesting tourist destinations. The ship has 1,000 cabins, so it is a challenge to fill the ship because of the high fares charged. In particular, the average
To help fill its seats for a particular flight, an airline offers a special nonrefundable fare of $100 for customers who make a reservation at least 21 days in advance and satisfy other restrictions. Thereafter, the fare will be $300. A total of 100 reservations will be accepted. The number of
Reconsider the Transcontinental Airlines example presented in Sec. 18.8. Management has concluded that the original estimate of $500 for the intangible cost of a loss of goodwill on the part of a bumped customer is much too low and should be increased to $1,000. Use the overbooking model to
The management of Quality Airlines has decided to base its overbooking policy on the overbooking model presented in Sec. 18.8. This policy now needs to be applied to a new flight from Seattle to Atlanta. The airplane has 125 seats available for a nonrefundable fare of $250. However, since there
The Blue Cab Company is the primary taxi company in the city of Maintown. It uses gasoline at the rate of 10,000 gallons per month. Because this is such a major cost, the company has made a special arrangement with the Amicable Petroleum Company to purchase a huge quantity of gasoline at a reduced
Consider the overbooking model presented in Sec. 18.8. For a specific application, suppose that the parameters of the model are p = 0.5, r = $1,000, s = $5,000, and L = 3. Use the binomial distribution directly (not the normal approximation) to calculate n*, the optimal number of reservations to
The Mountain Top Hotel is a luxury hotel in a popular ski resort area. The hotel always is essentially full during winter months, so reservations and payments must be made months in advance for week-long stays from Saturday to Saturday. Reservations can be canceled until a month in advance but are
Robert Gates rounds the corner of the street and smiles when he sees his wife pruning rose bushes in their front yard. He slowly pulls his car into the driveway, turns off the engine, and falls into his wife’s open arms. “How was your day?” she asks. “Great! The drugstore business could not
A young entrepreneur will be operating a firecracker stand for the Fourth of July. He has time to place only one order for the firecrackers he will sell from his stand. After obtaining the relevant financial data and some information with which to estimate the probability distribution of potential
American Aerospace produces military jet engines. Frequent shortages of one critical part has been causing delays in the production of the most popular jet engine, so a new inventory policy needs to be developed for this part. There is a long lead time between when an order is placed for the part
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 18.7. Briefly describe how inventory theory was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Freddie the newsboy runs a newstand. Because of a nearby financial services office, one of the newspapers he sells is the daily Financial Journal. He purchases copies of this newspaper from its distributor at the beginning of each day for $1.50 per copy, sells it for $2.50 each, and then receives a
Jennifer€™s Donut House serves a large variety of doughnuts, one of which is a blueberry-filled, chocolate-covered, supersized doughnut supreme with sprinkles. This is an extra large doughnut that is meant to be shared by a whole family. Since the dough requires so long to rise,
Swanson’s Bakery is well known for producing the best fresh bread in the city, so the sales are very substantial. The daily demand for its fresh bread has a uniform distribution between 300 and 600 loaves. The bread is baked in the early morning, before the bakery opens for business, at a cost of
Reconsider Prob. 18.7-5. The bakery owner, Ken Swanson, now wants you to conduct a financial analysis of various inventory policies. You are to begin with the policy obtained in the first four parts of Prob. 18.7-5 (ignoring any cost for the loss of customer goodwill). As given with the answers in
For the basic EOQ model, use the square root formula to determine how Q* would change for each of the following changes in the costs or the demand rate. (Unless otherwise noted, consider each change by itself.) (a) The setup cost is reduced to 25 percent of its original value. (b) The annual demand
Reconsider Prob. 18.7-5. The bakery owner, Ken Swanson, now has developed a new plan to decrease the size of shortages. The bread will be baked twice a day, once before the bakery opens (as before) and the other during the day after it becomes clearer what the demand for that day will be. The first
Suppose that the demand D for a spare airplane part has an exponential distribution with mean 50, that is,This airplane will be obsolete in 1 year, so all production of the spare part is to take place at present. The production costs now are $1,000 per item€”that is, c =
Reconsider Prob. 18.6-1 involving Henry Edsel’s car dealership. The current model year is almost over, but the Tritons are selling so well that the current inventory will be depleted before the end-of-year demand can be satisfied. Fortunately, there still is time to place one more order with the
Find the optimal ordering policy for the stochastic single-period model with a setup cost where the demand has the probability density functionand the costs are Holding cost = $1 per item, Shortage cost = $3 per item, Setup cost = $1.50, Production cost = $2 per item. Show your work, and then check
Using the approximation for finding the optimal policy for the stochastic single-period model with a setup cost when demand has an exponential distribution, find this policy whenand the costs are Holding cost = 40 cents per item, Shortage cost = $1.50 per item, Purchase price = $1 per item, Setup
Kris Lee, the owner and manager of the Quality Hardware Store, is reassessing his inventory policy for hammers. He sells an average of 50 hammers per month, so he has been placing an order to purchase 50 hammers from a wholesaler at a cost of $20 per hammer at the end of each month. However, Kris
Consider Example 1 (manufacturing speakers for TV sets) introduced in Sec. 18.1 and used in Sec. 18.3 to illustrate the EOQ models. Use the EOQ model with planned shortages to solve this example when the unit shortage cost is changed to $5 per speaker short per month.
Speedy Wheels is a wholesale distributor of bicycles. Its Inventory Manager, Ricky Sapolo, is currently reviewing the inventory policy for one popular model that is selling at the rate of 500 per month. The administrative cost for placing an order for this model from the manufacturer is $1,000 and
Reconsider Prob. 18.3-3. Because of the popularity of the Power model computer, Tim Madsen has found that customers are willing to purchase a computer even when none are currently in stock as long as they can be assured that their order will be filled in a reasonable period of time. Therefore, Tim
Every Saturday night a man plays poker at his home with the same group of friends. If he provides refreshments for the group (at an expected cost of $14) on any given Saturday night, the group will begin the following Saturday night in a good mood with probability 7/8 and in a bad mood with
When a tennis player serves, he gets two chances to serve in bounds. If he fails to do so twice, he loses the point. If he attempts to serve an ace, he serves in bounds with probability 3/8. If he serves a lob, he serves in bounds with probability 7/8. If he serves an ace in bounds, he wins the
Each year Ms. Fontanez has the chance to invest in two different no-load mutual funds: the Go-Go Fund or the Go-Slow Mutual Fund. At the end of each year, Ms. Fontanez liquidates her holdings, takes her profits, and then reinvests. The yearly profits of the mutual funds depend on where the market
Buck and Bill Bogus are twin brothers who work at a gas station and have a counterfeiting business on the side. Each day a decision is made as to which brother will go to work at the gas station, and then the other will stay home and run the printing press in the basement. Each day that the machine
Consider an infinite-period inventory problem involving a single product where, at the beginning of each period, a decision must be made about how many items to produce during that period. The setup cost is $10, and the unit production cost is $5. The holding cost for each item not sold during the
Reconsider Prob. 19.2-2. (a) Formulate a linear programming model for finding an optimal policy.
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 19.2. Briefly describe how Markov decision processes were applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Reconsider Prob. 19.2-3. (a) Formulate a linear programming model for finding an optimal policy.
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