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introduction to operations research
Introduction To Operations Research 7th Edition Frederick S. Hillier, Gerald J. Lieberman - Solutions
Reconsider Prob. 21.2-6.(a) Formulate a linear programming model for finding an optimal policy.C (b) Use the simplex method to solve this model. Use the resulting optimal solution to identify an optimal policy.
Reconsider Prob. 21.2-5.(a) Formulate a linear programming model for finding an optimal policy.C (b) Use the simplex method to solve this model. Use the resulting optimal solution to identify an optimal policy.
Reconsider Prob. 21.2-4.(a) Formulate a linear programming model for finding an optimal policy.C (b) Use the simplex method to solve this model. Use the resulting optimal solution to identify an optimal policy.
Reconsider Prob. 21.2-3.(a) Formulate a linear programming model for finding an optimal policy.C (b) Use the simplex method to solve this model. Use the resulting optimal solution to identify an optimal policy.
Reconsider Prob. 21.2-2.(a) Formulate a linear programming model for finding an optimal policy.C (b) Use the simplex method to solve this model. Use the resulting optimal solution to identify an optimal policy.
Reconsider Prob. 21.2-1.(a) Formulate a linear programming model for finding an optimal policy.C (b) Use the simplex method to solve this model. Use the resulting optimal solution to identify an optimal policy.
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
A person often finds that she is up to 1 hour late for work.If she is from 1 to 30 minutes late, $4 is deducted from her paycheck; if she is from 31 to 60 minutes late for work, $8 is deducted from her paycheck. If she drives to work at her normal speed (which is well under the speed limit), she
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
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 are dependent upon how the
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 38. If he serves a lob, he serves in bounds with probability 78. If he serves an ace in bounds, he wins the point
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 78 and in a bad mood with
A soap company specializes in a luxury type of bath soap.The sales of this soap fluctuate between two levels—low and high—depending upon two factors: (1) whether they advertise and(2) the advertising and marketing of new products by competitors.The second factor is out of the company’s
A student is concerned about her car and does not like dents. When she drives to school, she has a choice of parking it on the street in one space, parking it on the street and taking up two spaces, or parking in the lot. If she parks on the street in one space, her car gets dented with probability
During any period, a potential customer arrives at a certain facility with probability 12. If there are already two people at the facility (including the one being served), the potential customer leaves the facility immediately and never returns. However, if there is one person or less, he enters
Quality Bikes is a wholesale firm that specializes in the distribution of bicycles. In the past, the company has maintained ample inventories of bicycles to enable filling orders immediately, so informal rough forecasts of demand were sufficient to make the decisions on when to replenish inventory.
Reconsider Prob. 20.7-12.The economy is beginning to boom so the management of Transcontinental Airlines is predicting that the number of people flying will steadily increase this year over the relatively flat (seasonally adjusted) level of last year. Since the forecasting methods considered in
Transcontinental Airlines maintains a computerized forecasting system to forecast the number of customers in each fare class who will fly on each flight in order to allocate the available reservations to fare classes properly. For example, consider economy-class customers flying in midweek on the
Reconsider Prob. 20.5-3.Ralph Billett realizes that the last-value method is considered to be the naive forecasting method, so he wonders whether he should be using another method. Therefore, he has decided to use the available Excel templates that consider seasonal effects to apply various
The Centerville Water Department provides water for the entire town and outlying areas. The number of acre-feet of water consumed in each of the four seasons of the three preceding years is shown below T (a) Determine the seasonal factors for the four seasons.T (b) After considering seasonal
Follow the instructions of Prob. 20.7-15 for a product with the following sales history.
Management of the Jackson Manufacturing Corporation wishes to choose a statistical forecasting method for forecasting total sales for the corporation. Total sales (in millions of dollars)for each month of last year are shown below.(a) Note how the sales level is shifting significantly from month to
Reconsider Prob. 20.7-4.Ben Swanson now has decided to use the exponential smoothing method to forecast future sales of washing machines, but he needs to decide on which smoothing constant to use. Using an initial estimate of 24, apply this method retrospectively to the 12 months of last year with
Ben Swanson, owner and manager of Swanson’s Department Store, has decided to use statistical forecasting to get a better handle on the demand for his major products. However, Ben now needs to decide which forecasting method is most appropriate for each category of product. One category is major
The Healthwise Company produces a variety of exercise equipment. Healthwise management is very pleased with the increasing sales of its newest model of exercise bicycle. The sales during the last two months have been 4,655 and then 4,935.Management has been using exponential smoothing with trend,
A manufacturer sells a certain product in batches of 100 to wholesalers. The following table shows the quarterly sales figure for this product over the last several years.The company incorporates seasonal effects into its forecasting of future sales. It then uses exponential smoothing (with
Figure 20.3 shows CCW’s average daily call volume for each quarter of the past three years, and column F of Fig. 20.4 gives the seasonally adjusted call volumes. Management now wonders what these seasonally adjusted call volumes would have been if the company had started using seasonal factors
A company uses exponential smoothing with 1 2 to forecast demand for a product. For each month, the company keeps a record of the forecast demand (made at the end of the preceding month) and the actual demand. Some of the records have been lost;the remaining data appear in the table below.(a)
After graduating from college with a degree in mathematical statistics, Ann Preston has been hired by the Monty Ward Company to apply statistical methods for forecasting the company’s sales. For one of the company’s products, the moving-average method based upon sales in the 10 most recent
Select three of the applications of statistical forecasting methods listed in Table 20.1. Read the articles describing the applications in the indicated issues of Interfaces. For each one, write a one-page summary of the application and the benefits it provided.
Select one of the applications of statistical forecasting methods listed in Table 20.1. Read the article describing the application in the indicated issue of Interfaces. Write a two-page summary of the application and the benefits it provided.
For the infinite-period model with no setup cost, show that the value of y0 that satisfies(y0) p p c(1 h )is equivalent to the value of y that satisfiesdL d(y y) c(1 ) 0, where L(y), the expected shortage plus holding cost, is given by L(y)y p( y)D() d y 0h(y )D() d
Find the optimal (k, Q) policy for Prob. 19.7-10 for an infinite-period model with a discount factor of 0.90.
Consider a one-period model where the only two costs are the holding cost, given by h(y D) 1 30(y D), for y D, and the shortage cost, given by p(D y) 2.5(D y), for D y.The probability density function for demand is given byD()If you order, you must order an integer number of batches
The weekly demand for a certain type of electronic calculator is estimated to be The unit cost of these calculators is $80. The holding cost is $0.70 per calculator remaining at the end of a week. The shortage cost is $2 per calculator of unsatisfied demand at the end of a week.Using a weekly
A supplier of high-fidelity receiver kits is interested in using an optimal inventory policy. The distribution of demand per month is uniform between 2,000 and 3,000 kits. The supplier’s cost for each kit is $150. The holding cost is estimated to be $2 per kit remaining at the end of a month, and
Solve the inventory problem given in Prob. 19.7-6, but assume that the policy is to be used for only 1 year (a 12-period model). Shortages are backlogged each month, except that any shortages remaining at the end of the year are made up by purchasing similar items at a unit cost of $2. Any
Reconsider Prob. 20.7-9.Despite some fluctuations from year to year, note that there has been a basic trend upward in the annual demand for copper ore over the past 10 years. Therefore, by projecting this trend forward, causal forecasting can be used to forecast demands in future years by letting
Determine the optimal inventory policy when the goods are to be ordered at the end of every month from now on. The cost of bringing the inventory level up to y when x already is available is given by 2(y x). Similarly, the cost of having the monthly demand D exceed y is given by 5(D y). The
Solve Prob. 19.7-3 for an infinite-period model.
Solve Prob. 19.7-3 for a two-period model, assuming no salvage value, no backlogging at the end of the second period, and no discounting.
Find the optimal inventory policy for the following two-period model by using a discount factor of 0.9. The demand D has the probability density functionD()for 0 otherwise,and the costs are Holding cost $0.25 per item, Shortage cost $2 per item, Purchase price $1 per item.Stock left over at
Consider the following inventory situation. Demands in different periods are independent but with a common probability density function D() 5 10 for 0 50. Orders may be placed at the start of each period without setup cost at a unit cost of c 10. There are a holding cost of 8 per unit
Consider the following inventory situation. Demands in different periods are independent but with a common probability density function given byD()Orders may be placed at the start of each period without setup cost at a unit cost of c 10. There are a holding cost of 6 per unit remaining in stock
Consider the following inventory model, which is a single-period model with known density of demand D() e for 0 and D() 0 elsewhere. There are two costs connected with the model. The first is the purchase cost, given by c( y x).The second is a cost p that is incurred once if there is any
The campus bookstore must decide how many textbooks to order for a course that will be offered only once. The number of students who will take the course is a random variable D, whose distribution can be approximated by a (continuous) uniform distribution on the interval [40, 60]. After the quarter
The following data relate road width x and accident frequency y. Road width (in feet) was treated as the independent variable, and values y of the random variable Y, in accidents per 108 vehicle miles, were observed.Assume that Y is normally distributed with mean A Bx and constant variance for all
The management of Quality Airlines has decided to base its overbooking policy on the stochastic single-period model for perishable products, since this will maximize expected profit. This policy now needs to be applied to a new flight from Seattle to Atlanta. The airplane has 125 seats available
A college student, Stan Ford, recently took a course in operations research. He now enjoys applying what he learned to optimize his personal decisions. He is analyzing one such decision currently, namely, how much money (if any) to take out of his savings account to buy $100 traveler’s checks
Suppose that the demand D for a spare airplane part has an exponential distribution with mean 50, that is,5 10e /50 for 0D()0 otherwise.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
Reconsider Prob. 19.6-4.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
The following data are observations yi on a dependent random variable Y taken at various levels of an independent variable x. [It is assumed that E(Yixi) A Bxi, and the Yi are independent normal random variables with mean 0 and variance 2.](a) Estimate the linear relationship by the method of
A newspaper stand purchases newspapers for $0.36 and sells them for $0.50. The shortage cost is $0.50 per newspaper (because the dealer buys papers at retail price to satisfy shortages).The holding cost is $0.002 per newspaper left at the end of the day.The demand distribution is a uniform
Micro-Apple is a manufacturer of personal computers. It currently manufactures a single model—the MacinDOS—on an assembly line at a steady rate of 500 per week. MicroApple orders the floppy disk drives for the MacinDOS (1 per computer) from an outside supplier at a cost of $30 each. Additional
When using the stochastic continuous-review model presented in Sec. 19.5, 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
If a particle is dropped at time t 0, physical theory indicates that the relationship between the distance traveled r and the time elapsed t is r gt k for some positive constants g and k. A transformation to linearity can be obtained by taking logarithms:log r log g k log t.By letting y log r, A
Consider a situation where a particular product is produced and placed in in-process inventory until it is needed in a subsequent production process. No units currently are in inventory, but three units will be needed in the coming month and an additional four units will be needed in the following
MBI is a manufacturer of personal computers. All its personal computers use a 3.5-inch high-density floppy disk drive which it purchases from Ynos. MBI operates its factory 52 weeks per year, which requires assembling 100 of these floppy disk drives into computers per week. MBI’s annual holding
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 ab. Replenishments and withdrawals of
Consider the EOQ model with planned shortages, as presented in Sec. 19.3. Suppose, however, that the constraint S/Q 0.8 is added to the model. Derive the expression for the optimal value of Q.
Cindy Stewart and Misty Whitworth graduated from business school together. They now are inventory managers for competing wholesale distributors, making use of the scientific inventory management techniques they learned in school. Both of them are purchasing 85-horsepower speedboat engines for their
Computronics is a manufacturer of calculators, currently producing 200 per week. One component for every calculator is a liquid crystal display (LCD), which the company purchases from Displays, Inc. (DI) for $1 per LCD. Computronics management wants to avoid any shortage of LCDs, since this would
The Becker Company factory has been experiencing long delays in jobs going through the turret lathe department because of inadequate capacity. The head of this department contends that five machines are required, as opposed to the three machines now in place. However, because of pressure from
Reconsider Prob. 17.6-33.(a) Formulate part (a) to fit as closely as possible a special case of one of the decision models presented in Sec. 18.4. (Do not solve.)(b) Describe Alternatives 2 and 3 in queueing theory terms, including their relationship (if any) to the decision models presented in
Greg is making plans to open a new fast-food restaurant soon. He is estimating that customers will arrive randomly (a Poisson process) at a mean rate of 150 per hour during the busiest times of the day. He is planning to have three employees directly serving the customers. He now needs to make a
Consider the special case of model 2 where (1) any /s is feasible and (2) both f() and the waiting-cost function are linear functions, so that E(TC) Crs CwL,where Cr is the marginal cost per unit time for each unit of a server’s mean service rate and Cw is the cost of waiting per unit time
An airline maintenance base wants to make a change in its overhaul operation. The present situation is that only one airplane can be repaired at a time, and the expected repair time is 36 hours, whereas the expected time between arrivals is 45 hours. This situation has led to frequent and prolonged
Jerry Jansen, Materials Handling Manager at the CasperEdison Corporation’s new factory, needs to make a purchasing decision. He needs to choose between two types of materialshandling equipment, a small tractor-trailer train and a heavy-duty forklift truck, for transporting heavy goods between
Customers arrive at a fast-food restaurant with one server according to a Poisson process at a mean rate of 30 per hour. The server has just resigned, and the two candidates for the replacement are X (fast but expensive) and Y (slow but inexpensive). Both candidates would have an exponential
A certain queueing system has a Poisson input, with a mean arrival rate of 4 customers per hour. The service-time distribution is exponential, with a mean of 0.2 hour. The marginal cost of providing each server is $20 per hour, where it is estimated that the cost that is incurred by having each
Section 18.3 indicates that a linear waiting-cost function yields E(WC) CwL, where Cw is the cost of waiting per unit time for each customer. In this case, the objective for decision model 1 in Sec. 18.4 is to minimize E(TC) Css CwL. The purpose of this problem is to enable you to explore the
Follow the instructions of Prob. 18.3-1 for the following waiting-cost functions.(a) g(N)(b) h()
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.(b) Find the steady-state distribution of the number of customers at facility 1, facility 2, and facility 3. Then show the product form
Consider a system of two infinite queues in series, where each of the two service facilities has a single server. All service times are independent and have an exponential distribution, with a mean of 3 minutes at facility 1 and 4 minutes at facility 2. Facility 1 has a Poisson input process with a
Consider a queueing system with two servers, where the customers arrive from two different sources. From source 1, the customers always arrive 2 at a time, where the time between consecutive arrivals of pairs of customers has an exponential distribution with a mean of 20 minutes. Source 2 is itself
Consider the finite queue variation of the M/G/1 model, where K is the maximum number of customers allowed in the system. For n 1, 2, . . . , let the random variable Xn be the number of customers in the system at the moment tn when the nth customer has just finished being served. (Do not count the
A company has one repair technician to keep a large group of machines in running order. Treating this group as an infinite calling population, individual breakdowns occur according to a Poisson process at a mean rate of 1 per hour. For each breakdown, the probability is 0.9 that only a minor repair
Consider the E2/M/1 model with 4 and 5. This model can be formulated as a continuous time Markov chain by dividing each interarrival time into two consecutive phases, each having an exponential distribution with a mean of 1/(2) 0.125, and then defining the state of the system as (n, p), where
Consider a single-server queueing system with a Poisson input, Erlang service times, and a finite queue. In particular, suppose that k 2, the mean arrival rate is 2 customers per hour, the expected service time is 0.25 hour, and the maximum permissible number of customers in the system is 2. This
Consider a queueing system with a Poisson input, where the server must perform two distinguishable tasks in sequence for each customer, so the total service time is the sum of the two task times (which are statistically independent).(a) Suppose that the first task time has an exponential
Reconsider Prob. 17.7-6.Management has adopted the proposal but now wants further analysis conducted of this new queueing system.(a) How should the state of the system be defined in order to formulate the queueing model as a continuous time Markov chain?(b) Construct the corresponding rate diagram.
For the state-dependent model presented at the end of Sec. 17.6, show the effect of the pressure coefficient c by using Fig. 17.10 to construct a table giving the ratio (expressed as a decimal number) of L for this model to L for the corresponding M/M/s model (i.e., with c 0). Tabulate these ratios
Consider a single-server queueing system. It has been observed that (1) this server seems to speed up as the number of customers in the system increases and (2) the pattern of acceleration seems to fit the state-dependent model presented at the end of Sec.17.6. Furthermore, it is estimated that the
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
For the finite queue variation of the M/M/1 model, develop an expression analogous to Eq. (1) in Prob. 17.6-17 for the following probabilities:(a) P{ t}.(b) P{q t}.[Hint: Arrivals can occur only when the system is not full, so the probability that a random arrival finds n customers already there is
The reservation office for Central Airlines has two agents answering incoming phone calls for flight reservations. In addition, one caller can be put on hold until one of the agents is available to take the call. If all three phone lines (both agent lines and the hold line) are busy, a potential
Consider a generalization of the M/M/1 model where the server needs to “warm up” at the beginning of a busy period, and so serves the first customer of a busy period at a slower rate than other customers. In particular, if an arriving customer finds the server idle, the customer experiences a
You are given an M/M/1 queueing system in which the expected waiting time and expected number in the system are 120 minutes and 8 customers, respectively. Determine the probability that a customer’s service time exceeds 20 minutes.
Derive Wq directly for the following cases by developing and reducing an expression analogous to Eq. (1) in Prob. 17.6-17.(Hint: Use the conditional expected waiting time in the queue given that a random arrival finds n customers already in the system.)(a) The M/M/1 model (b) The M/M/s model
Section 17.6 gives the following equations for the M/M/1 model:(1) P{ t}n0 PnP{Sn1 t}.(2) P{ t} e(1)t.Show that Eq. (1) reduces algebraically to Eq. (2). (Hint: Use differentiation, algebra, and integration.)
A gas station with only one gas pump employs the following policy: If a customer has to wait, the price is $1 per gallon; if she does not have to wait, the price is $1.20 per gallon. Customers arrive according to a Poisson process with a mean rate of 15 per hour. Service times at the pump have an
Airplanes arrive for takeoff at the runway of an airport according to a Poisson process at a mean rate of 20 per hour. The time required for an airplane to take off has an exponential distribution with a mean of 2 minutes, and this process must be completed before the next airplane can begin to
Consider the M/M/s model with a mean arrival rate of 10 customers per hour and an expected service time of 5 minutes.Use the Excel template for this model to obtain and print out the various measures of performance (with t 10 and t 0, respectively, for the two waiting time probabilities) when the
For each of the following statements about an M/M/1 queueing system, label the statement as true or false and then justify your answer by referring to specific statements (with page citations) in the chapter.(a) The waiting time in the system has an exponential distribution.(b) The waiting time in
Consider the following statements about an M/M/1 queueing system and its utilization factor . Label each of the statements as true or false, and then justify your answer.(a) The probability that a customer has to wait before service begins is proportional to .(b) The expected number of customers
Verify the following relationships for an M/M/1 queueing system: 17.6-5.It is necessary to determine how much in-process storage space to allocate to a particular work center in a new factory.Jobs arrive at this work center according to a Poisson process with a mean rate of 3 per hour, and the time
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
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
The Copy Shop is open 5 days per week for copying materials that are brought to the shop. It has three identical copying machines that are run by employees of the shop. Only two operators are kept on duty to run the machines, so the third machine is a spare that is used only when one of the other
Consider a self-service model in which the customer is also the server. Note that this corresponds to having an infinite number of servers available. Customers arrive according to a Poisson process with parameter , and service times have an exponential distribution with parameter .(a) Find Lq and
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