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business
operations research an introduction
Questions and Answers of
Operations Research An Introduction
In Example 19.1-1, estimate the area of the circle using the first two columns of the 0-1 random numbers in Table 19.1. (For convenience, go down each column, selecting R1 first and then R2.) How
In the cost model in Section 18.9.1, it is generally difficult to estimate the cost parameter C2 (cost of waiting). As a result, it may be helpful to compute the cost C2 implied by the aspiration
A shop uses 10 identical machines. Each machine breaks down once every 7 hrs on the average. It takes half an hour on the average to repair a broken machine. Both the breakdown and repair processes
The necessary conditions for ETC(c) (defined earlier) to assume a minimum value at c = c* are ETC1c* - 12 Ú ETC1c*2 and ETC1c* + 12 Ú ETC1c*2 Show that these conditions reduce to Ls1c*2 - Ls1c* +
A machine shop includes 20 machines and 3 repairpersons. A working machine breaks down randomly according to a Poisson distribution. The repair time per machine is exponential with a mean of 6
A company leases a wide-area telecommunications service (WATS) telephone line for $2000 a month. The office is open 200 working hours per month. At all other times, the WATS line service is used for
Tasco Oil owns a pipeline booster unit that operates continuously. The time between breakdowns for each booster is exponential with a mean of 20 hrs. The repair time is exponential with mean 3 hrs.
Solve Example 18.9-2, assuming that C1 = $25 and C2 = $50.
Second Time Around sells popular used items on consignment. Its operation can be viewed as an inventory problem in which the stock is replenished and depleted randomly according to Poisson
Suppose in Problem 18-118 that the investor can choose any desired restaurant capacity based on a specific marginal cost for each additional capacity unit requested. Derive the associated general
Pizza Unlimited sells two franchised restaurant models. Model A has a capacity of 20 groups of customers, and model B can seat 30 groups. The monthly cost of operating model A is $12,000 and that of
H&I Industry produces a special machine with different production rates (pieces per hour) to meet customer specifications. A shop owner is considering buying one of these machines and wants to decide
BB&K Groceries is opening a new store boasting “state-of-the-art” check-out scanners. Mr. Bih, one of the owners of B&K, has limited the choices to two scanners:scanner A can process 15 items a
Metalco is in the process of hiring a repairperson for a 10-machine shop. Two candidates are under consideration. The first candidate can carry out repairs at the rate of 5 machines per hour and
In Example 18.9-1, do the following:(a) Verify the values of m2, m3, and m4 given in the example.(b) Suppose that the penalty of $48 per job per day is levied only on jobs that are not“in
In a service facility with c parallel servers, suppose that customers arrive according to a Poisson distribution, with a mean rate of l. Arriving customers are assigned to servers(busy or free) on a
Show that the P–K formula reduces to Ls of the 1M/M/12:1GD//2 when the service time is exponential with a mean of 1m time units.
1M/Em/12:1GD//2. Given that the service time is Erlang with parameters m and m(i.e., E5t6 = mm and var5t6 = m m2), show that the P–K formula reduces to Ls = mr +m11 + m2r2 211 - mr2
1M/D/12:1GD//2. Show that for the case where the service time is constant, the P-K formula reduces to P L = p + 2(1 - p) = where and p == AEt).
A product arrives according to a Poisson distribution at the rate of one every 45 minutes.The product requires two tandem operations attended by one worker. The first operation uses a semiautomatic
Optica makes prescription glasses according to orders received from customers. Each worker is specialized in certain types of glasses. The company has been experiencing unusual delays in the
Layson Roofing Inc. installs shingle roofs on new and old homes in Arkansas.Prospective customers request the service randomly at the rate of 6 jobs per 30-day month and are placed on a waiting list
Solve Example 18.7-1, assuming that the service-time distribution is given as follows:*(a) Uniform between 8 and 20 minutes.(b) Normal with m = 10 minutes and s = 3 minutes.(c) Discrete with values
In Example 18.7-1, compute the percentage of time the facility is idle.
Verify the following results for the special case of one repairperson 1R = 12: Pn Po K!p" (K-n)!' = (1 + L = K - Po R K!p" (Kn)) (1 - Po) P
Show that the rate of breakdown in the shop can be computed from the formula leff = mR where R is the average number of busy repairpersons.
Verify the expression for pn for the (M/M/R): (GD/K/K) model.
After a long wait, the Newborns were rewarded with quintuplets, two boys and three girls, thanks to the wonders of new medical advances. During the first 5 months, the babies’ life consisted of two
Kleen All is a service company that performs a variety of odd jobs, such as yard work, tree pruning, and house painting. The company’s four employees leave the office with the first assignment of
An operator attends five automatic machines. After each machine completes a batch run, the operator must reset it before a new batch is started. The time to complete a batch run is exponential with
In the computations in Figure 18.9, it may appear confusing that the average rate of machine breakdown in the shop, leff, increases with the increase in R. Explain why the increase in leff should be
In Example 18.6-8, define and compute the productivity of the repairpersons for R = 1, 2, 3, and 4. Use this information in conjunction with the measure of machine productivity to decide on the
In Example 18.6-8, do the following:(a) Verify the values of leff given in Figure 18.9.*(b) Compute the expected number of idle repairpersons, given R = 4.(c) Compute the probability that all
Repeat Problem 18-93 for large r = 9, and show that the same conclusion holds except that the value of c must be higher (at least 14). From the results of Problems 18-93 and 18-94, what general
Demonstrate (by using excelPoissonQ.xls or TORA) that for small r = .1, the values of Ls, Lq, Ws, Wq, and pn for c as small as 4 servers, the 1M/M/c2:1GD//2 model can be estimated reliably using
New drivers are required to pass written tests before they are given road driving test. These tests are usually administered in the city hall. Records at the City of Springdale show that the average
In Example 18.6-7, compute the following:(a) The probability that the investor will sell out completely.(b) The probability that the investor will own at least 20 securities.(c) The probability that
For 1M/M/c2:1GD/N/2 with which N =c, define ln and mn in terms of the general model (Section 18.5), then show that the expression for pn is given as pn =rn n!p0, n = 1, 2,c, c where p0 = a1 + a cn=1
Verify the expression for p0 and Lq for 1M/M/c2:1GD/N/2 when rc= 1.
Prove the following equality for 1M/M/c2:1GD/N/2:leff = mc, where c is the number of busy servers.
Verify the expression for p0 for the 1M/M/c2:1GD/N/2 model, given that rc 1.
At U of A, newly enrolled freshmen students are notorious for wanting to drive their cars to class (even though most of them are required to live on campus and can conveniently make use of the
A small engine repair shop is run by three mechanics. Early in March of each year, people bring in their tillers and lawn mowers for service and maintenance. The shop is willing to accept all the
Eat & Gas convenience store operates a two-pump gas station. The lane leading to the pumps can house at most 3 cars, excluding those being serviced. Arriving cars go elsewhere if the lane is full.
In Example 18.6-6, determine the following:(a) The expected number of idle cabs.(b) The probability that a calling customer will be next to last on the list.(c) The limit on the waiting list if it is
For the 1M/M/c2:1GD//2 model, show the following:(a) The probability that a customer is waiting is r 1c - r2 pc.(b) The average number in the queue given that it is not empty is c 1c - r2 .(c) The
Show that for the 1M/M/c2:1GD//2 model Lq =cr 1c - r22 pc
Show that pn for the 1M/M/12:1GD//2 model can be obtained from that of the 1M/M/c2:1GD//2 model by setting c = 1.
Prove that Ls = Lq + c starting with the definition Lq = gn = c + 11n - C2pn, where c is the average number of busy servers. Hence, show that c = leff m .
In the derivation of pn for the 1M/M/c2:1GD//2 model, indicate which part of the derivation requires the condition rc 6 1. Explain verbally the meaning of the condition.What will happen if the
For the 1M/M/c2:1GD//2 model, Morse (1958, p. 103) shows that as rc S 1, Lq =r c - r Noting that rc S 1 means that the servers are extremely busy, use this information to show that the ratio of the
In the United States, the use of single-line, multiple-server queues is common in post offices and in passenger check-in counters at airports. However, both grocery stores and banks (especially in
Drake Airport services rural, suburban, and transit passengers. The arrival distribution for each of the three groups is Poisson with mean rates of 15, 10, and 20 passengers per hour, respectively.
The U of A computer center is equipped with four identical mainframe computers. The number of users at any time is 25. Each user is capable of submitting a job from a terminal every 15 minutes, on
A small post office has two open windows. Customers arrive according to a Poisson distribution at the rate of 1 every 3 minutes. However, only 80% of them seek service at the windows. The service
McBurger fast-food restaurant has 3 cashiers. Customers arrive according to a Poisson distribution every 3 minutes and form one line to be served by the first available cashier. The time to fill an
Customers arrive at Thrift Bank according to a Poisson distribution, with a mean of 45 customers per hour. Transactions per customer last about 5 minutes and are exponentially distributed. The bank
Determine the minimum number of parallel servers needed in each of the following(Poisson arrival/departure) situations to guarantee that the operation of the queuing situation will be stable (i.e.,
In the cab company example, suppose that the average time per ride is actually about 14.5 minutes, so that the utilization 1= l mc2 for the 2- and 4-cab operations increases to more than 96%. Is it
Consider Example 18.6-5.(a) Show that the remarkable reduction in waiting time by more than 50% for the consolidated case is coupled with an increase in the percentage of time the servers remain
Show that leff for 1M/M/12:1GD/N/2 can be computed from the formula leff = m1Ls - Lq2
Show that when r = 1 for 1M/M/12:1GD/N/2, the expected number in the system, Ls, equals N2. 1Hint: 1 + 2 + c + i = i1i + 12 2 .2
The probabilities pn of n customers in the system for an (M/M/1):1GD/5/2 are given in the following table:n 0 1 2 3 4 5 pn .399 .249 .156 .097 .061 .038 The arrival rate l is five customers per
Patients arrive at a 1-doctor clinic according to a Poisson distribution at the rate of 20 patients per hour. The waiting room does not accommodate more than 14 patients.Examination time per patient
A cafeteria can seat a maximum of 50 persons. Customers arrive in a Poisson stream at the rate of 10 per hour and are served (one at a time) at the rate of 12 per hour.(a) What is the probability
The final assembly of electric generators at Electro is produced at the Poisson rate of 10 generators per hour. The generators are then conveyed on a belt to the inspection department for final
The time barber Joe takes to give a haircut is exponential with a mean of 12 minutes.Because of his popularity, customers usually arrive (according to a Poisson distribution) at a rate much higher
Consider the car wash facility of Example 18.6-4. Determine the number of parking spaces such that the percentage of cars that cannot find a space does not exceed 3%.
In Example 18.6-4, determine the following:(a) Probability that an arriving car will go into the wash bay immediately on arrival.(b) Expected waiting time until a service starts.(c) Expected number
For the 1M/M/12:1GD//2, show that(a) The expected number in the queue, given that the queue is not empty, = 1 11 - r2 .(b) The expected waiting time in the queue for those who must wait = 1 1 m -
For the 1M/M/12:1GD//2, derive the expression for Lq using the basic definition gn = 21n - 12pn.
In the 1M/M/12:1GD//2, give a plausible argument as to why Ls does not equal Lq + 1, in general. Under what condition will the equality hold?
Customers arrive at a one-window drive-in bank according to a Poisson distribution, with a mean of 10 per hour. The service time per customer is exponential, with a mean of 5 minutes. There are three
A fast-food restaurant has one drive-in window. Cars arrive according to a Poisson distribution at the rate of 2 cars every 5 minutes. The space in front of the window can accommodate at most 10
Cars arrive at the Lincoln Tunnel toll gate according to a Poisson distribution, with a mean of 90 cars per hour. The time for passing the gate is exponential with mean 38 seconds.Drivers complain of
Over the years, Detective Columbo, of the Fayetteville Police Department, has had phenomenal success in solving every single crime case. It is only a matter of time before any case is solved. Columbo
John Macko is a student at Ozark U. He does odd jobs to supplement his income.Job requests come every 5 days on the average, but the time between requests is exponential. The time for completing a
In Example 18.6-2, do the following.(a) Determine the percent utilization of the wash bay.(b) Determine the probability that an arriving car must wait in the parking lot prior to entering the wash
Solve Example 18.6-1 using the following data: number of parking spaces = 6, number of temporary spaces = 4, l = 10 cars per hour, and average parking time = 45 minutes.
In Example 18.6-1, do the following:*(a) Compute Lq directly using the formula gn = c + 11n - c2pn.(b) Compute Ws from Lq.*(c) Compute the average number of cars that will not be able to enter the
The induction proof for deriving the general solution of the generalized model is applied as follows. ConsiderWe substitute for pn-1 and pn-2 in the general difference equation involving pn, pn-1,
Consider the single-queue model where only one customer is allowed in the system.Customers who arrive and find the facility busy never return. Assume that the arrivals distribution is Poisson with
Consider a one-server queuing situation in which the arrival and service rates are given by ln = 10 - n, n = 0, 1, 2, 3 mn =n 2 + 5, n = 1, 2, 3, 4 This situation is equivalent to reducing the
A barbershop serves one customer at a time and provides three seats for waiting customers. If the place is full, customers go elsewhere. Arrivals occur according to a Poisson distribution with mean
Have you ever heard someone repeat the contradictory statement, “The place is so crowded no one goes there any more”? This statement can be interpreted as saying that the opportunity for balking
First Bank of Springdale operates a one-lane drive-in ATM machine. Cars arrive according to a Poisson distribution at the rate of 10 cars per hour. The time per car needed to complete the ATM
In the B&K model of Example 18.5-1, suppose that all three counters are always open and that the operation is set up such that the customer will go to the first empty counter. Determine the
In the B&K model of Example 18.5-1, suppose that the interarrival time at the checkout area is exponential with mean 8 minutes and that the checkout time per customer is also exponential with mean 12
In Example 18.5-1, determine the following:(a) The probability distribution of the number of open counters.(b) The average number of busy counters.
Derive the truncated Poisson distribution from the difference-differential equations of the pure death model using induction. [Note: See the hint in Problem 18-28.]
Prove that the distribution of the time between departures corresponding to the truncated Poisson in the pure death model is an exponential distribution with mean 1m time units.
Demand for an item occurs according to a Poisson distribution with mean 3 per day.The maximum stock level is 25 items, which occurs on each Monday immediately after a new order is received. The order
A machine shop has just stocked 10 spare parts for the repair of a machine. Stock replenishment that brings the stock level back to 10 pieces occurs every 7 days. The time between breakdowns is
Inventory is withdrawn from a stock of 80 items according to a Poisson distribution at the rate of 5 items per day. Determine the following:(a) The probability that 10 items are withdrawn during the
A freshman student receives a bank deposit of $100 a month from home to cover incidentals. Withdrawal checks of $20 each occur randomly during the month and are spaced according to an exponential
Each morning, the refrigerator in a small machine shop is stocked with two cases (24 cans per case) of soft drinks for use by the shop’s 12 employees. The employees can quench their thirst at any
The Springdale High School band is performing a benefit concert in its new 400-seat auditorium. Local businesses buy the tickets in blocks of 5 and donate them to youth organizations. Tickets go on
Consider Example 18.4-2. In each of the following cases, first write the answer algebraically, and then use excelPoissonQ.xls or TORA to provide numerical answers.*(a) The probability that the stock
In Example 18.4-2, use excelPoissonQ.xls or TORA to compute pn172, n = 1, 2,c, 18, and then verify manually that these probabilities yield E5n 0 t = 7 0 6 = .664 dozen.
Derive the Poisson distribution from the difference-differential equations of the pure birth model. Hint: The solution of the general differential equation is y' + a(t)y = b(t) y = e=Jar) { [b(1)/(0)
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