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practical management science

Practical Management Science, Revised 3rd Edition Wayne L Winston, S. Christian Albright - Solutions

In Example 14.4 of Section 14.5, we examined whether an MM1 system with a single fast server is better or worse than an MMs system with several slow servers. Keeping the same inputs as in the example, use simulation to see whether you obtain the same type of results as with the analytical
Make any assumptions about the warm-up time and runtime you believe are appropriate. Try solving the problem with exponentially distributed copying times. Then try it with gamma-distributed copying times, where the standard deviation is 3.2 minutes. Do you get the same recommendation on which
Simulate the system in Problem
Given the model in the Multiserver Simulation.xlsm file, what unit cost parameters should be used if we are interested in “optimizing” the system? Choose representative inputs and unit costs, and then illustrate how to use the simulation outputs to estimate total system costs.
How long does it take to reach steady state? Use simulation, with the Multiserver Simulation.xlsm file, to experiment with the effect of warm-up time and runtime on the key outputs. For each of the following, assume a five-server system with a Poisson arrival rate of 1 per minute and
Use exponentially distributed service times and a warm-up period of 1 hour for each.d. Why might the use of a long warm-up time bias the results toward worse system behavior than would actually be experienced? If you could ask the programmer of the simulation to provide another option concerning
The Smalltown Credit Union experiences its greatest congestion on paydays from 11:30 A.M. until 1:00 P.M.During these rush periods, customers arrive according to a Poisson process at rate 2.1 per minute. The credit union employs 10 tellers for these rush periods, and each takes 4.7 minutes to
Using the arrival rates from the lunchtime rush example, it seems sensible to vary the number of servers so that more servers work during the busy hours. In particular, suppose management wants to have an average of three servers working (in parallel) in any half-hour period, but the number working
Note that the sum of these rates is the same as the sum of the rates in the example, so that we expect the same total number of arrivals, but now they are more concentrated in the noon to 1 P.M. hour. Compare the results with these arrival rates to the results in the example. Write a short report
In the lunchtime rush example, the arrival rate changed fairly gradually throughout the period of interest.Assume now that the arrival rate first increases and then decreases in a more abrupt manner. Specifically, replace the arrival rates in the example by the following: 15, 20, 70, 85, 30, and
In the lunchtime rush example, we assumed that the system starts empty and idle at 11 A.M. Assume now that the restaurant opens earlier than 11 A.M., but we are still interested only in the period from 11 A.M. to 2 P.M.How does the initial number of customers present at 11 A.M. affect the results?
The small mail-order firm Sea’s Beginning has one phone line. An average of 60 people per hour call in orders, and it takes an average of 1 minute to handle a call. Time between calls and time to handle calls are exponentially distributed. If the phone line is busy, Sea’s Beginning can put up
Two one-barber shops sit side by side in Dunkirk Square. Each shop can hold a maximum of 4 people, and any potential customer who finds a shop full will not wait for a haircut. Barber 1 charges $11 per haircut and takes an average of 15 minutes to complete a haircut.Barber 2 charges $7 per haircut
Consider the following two queueing systems.■ System 1: An MM1 system with arrival rate and service rate 3μ■ System 2: An MM3 system with arrival rate and each server working at rate μWhich system will have the smaller W and L?The following problems are optional. They are based on the
On average, 100 customers arrive per hour at Gotham City Bank. It takes a teller an average of 2 minutes to serve a customer. Interarrival and service times are exponentially distributed. The bank currently has 4 tellers working. The bank manager wants to compare the following two systems with
The manager of a bank wants to use an MMs queueing model to weigh the costs of extra tellers against the cost of having customers wait in line. The arrival rate is 60 customers per hour, and the average service time is 4 minutes. The cost of each teller is easy to gauge at the $8.50 per hour wage
Referring to Problem 18, suppose the airline wants to determine how many checkpoints to operate to minimize operating costs and delay costs over a 10-year period. Assume that the cost of delaying a passenger for one hour is $10 and that the airport is open every day for 16 hours per day. It costs
For the MM1 queueing model, why do the following results hold? (Hint: Remember that 1 is the mean service time. Then think how long a typical arrival must wait in the system or in the queue.)a. W (L 1)b. WQ L
A worker at the State Unemployment Office is responsible for processing a company’s forms when it opens for business. The worker can process an average of four forms per week. In 2006, an average of 1.8 companies per week submitted forms for processing, and the worker had a backlog of 0.45 week.
A bank is trying to determine which of two machines to rent for check processing. Machine 1 rents for$10,000 per year and processes 1000 checks per hour.Machine 2 rents for $15,000 per year and processes 1600 checks per hour. Assume that machines work 8 hours a day, 5 days a week, 50 weeks a year.
Consider an airport where taxis and customers arrive(exponential interarrival times) with respective rates of 1 and 2 per minute. No matter how many other taxis are present, a taxi will wait. If an arriving customer does not find a taxi, the customer immediately leaves.a. Model this system as an
The limited source model can often be used to approximate the behavior of a computer’s CPU ( central processing unit). Suppose that 20 terminals (assumed to always be busy) feed the CPU. After the CPU responds to a user, the user takes an average of 80 seconds before sending another request to
A laundromat has 5 washing machines. A typical machine breaks down once every five days. A repairman can repair a machine in an average of 2.5 days.Currently, three repairmen are on duty. The owner of the laundromat has the option of replacing them with a superworker, who can repair a machine in an
On average, 40 cars per hour are tempted to use the drive-through window at the Hot Dog King Restaurant.(We assume that interarrival times are exponentially distributed.) If a total of more than four cars are in line (including the car at the window), a car will not enter the line. It takes an
A service facility consists of 1 server who can serve an average of 2 customers per hour (service times are exponential).An average of 3 customers per hour arrive at the facility (interarrival times are assumed to be exponential). The system capacity is 3 customers:2 waiting and 1 being served.a.
On average, 100 customers arrive per hour at the Gotham City Bank. The average service time for each customer is 1 minute. Service times and interarrival times are exponentially distributed. The manager wants to ensure that no more than 1% of all customers will have to wait in line for more than 5
MacBurger’s is attempting to determine how many servers to have available during the breakfast shift. On average, 100 customers arrive per hour at the restaurant.Each server can handle an average of 50 customers per hour. A server costs $5 per hour, and the cost of a customer waiting in line for
In this problem, all interarrival and service times are exponentially distributed.a. At present, the finance department and the marketing department each has its own typists. Each typist can type 25 letters per day. Finance requires that an average of 20 letters per day be typed, and marketing
A small bank is trying to determine how many tellers to employ. The total cost of employing a teller is $100 per day, and a teller can serve an average of 60 customers per day. On average, 50 customers arrive per day at the bank, and both service times and interarrival times are exponentially
A supermarket is trying to decide how many cash registers to keep open. Suppose an average of 18 customers arrive each hour, and the average checkout time for a customer is 4 minutes. Interarrival times and service times are exponentially distributed, and the system can be modeled as an MMs
Each airline passenger and his luggage must be checked to determine whether he is carrying weapons onto the airplane. Suppose that at Gotham City Airport, 10 passengers per minute arrive, on average.Also, assume that interarrival times are exponentially distributed. To check passengers for weapons,
Expand the MMs Template.xlsm file so that the steadystate probability distribution of the number in the system is shown in tabular form and graphically. That is, enter values 0, 1, and so on (up to some upper limit you can choose) in the range from cell E12 down and copy the formula in cell F12
In the MMs model, where μ is the service rate per server, explain why μ is not the appropriate condition for steady state, but sμ is.
If you average these, what parameter of the MMs model are you estimating?Use these numbers to estimate the arrival rate .If instead these numbers were observed service times, what would their average be an estimate of, and what would the corresponding estimate of μ be?
Suppose that you observe a sequence of interarrival times, such as 1.2, 3.7, 4.2, 0.5, 8.2, 3.1, 1.7, 4.2, 0.7, 0.3, and 2.0. For example, 4.2 is the time between the arrivals of customers 2 and
For an MM1 queueing system, we know that L (μ ). Suppose that and μ are both doubled.How does L change? How does W change? How does WQ change? How does LQ change? (Remember the basic queueing relationships, including Little’s formula.)
Expand the MM1 Template.xlsx file so that the steady-state probability distribution of the number in the system is shown in tabular form and graphically.That is, enter values 0, 1, and so on (up to some upper limit you can choose) in the range from cell E11 down and copy the formula in cell F11
The MM1 Template.xlsx file is now set up so that when you enter any time value in cell H11, the formula in cell I11 gives the probability that the wait in queue will be greater than this amount of time. Suppose that you would like the information to go the other direction. That is, you would like
The MM1 Template.xlsx file is now set up so that you can enter any integer in cell E11 and the corresponding probability of that many in the system appears in cell F11. Change this setup so that columns E and F specify the distribution of the number in the queue rather than the system. That is, set
The Decision Sciences Department is trying to determine whether to rent a slow or a fast copier. The department believes that an employee’s time is worth$15 per hour. The slow copier rents for $4 per hour, and it takes an employee an average of 10 minutes to complete copying. The fast copier
A fast-food restaurant has one drive-through window.On average, 40 customers arrive per hour at the window. It takes an average of 1 minute to serve a customer. Assume that interarrival and service times are exponentially distributed.a. On average, how many customers are waiting in line?b. On
Consider a fast-food restaurant where customers enter at a rate of 75 per hour, and 3 servers are working.Customers wait in a single line and go, in FCFS fashion, to the first of the 3 servers who is available.Each server can serve 1 customer every 2 minutes on average. If you are standing at the
Consider a bank where potential customers arrive at rate of 60 customers per hour. However, because of limited space, 1 out of every 4 arriving customers finds the bank full and leaves immediately (without entering the bank). Suppose that the average number of customers waiting in line in the bank
Little’s formula applies to an entire queueing system or to a subsystem of a larger system. For example, consider a single-server system composed of two subsystems.The first subsystem is the waiting line, and the second is the service area, where service actually takes place. Let be the rate
Assume that parts arrive at a machining center at a rate of 60 parts per hour. The machining center is capable of processing 75 parts per hour—that is, the mean time to machine a part is 0.8 minute. If you are watching these parts exiting the machine center, what exit rate do you observe, 60 or
This estimates the probability P(X 1). Now find all random numbers that are greater than 4.Among these, find the fraction that are greater than 5.This estimates the probability P(X 4 1|X 4).According to the memoryless property, these two estimates should be nearly equal. Are they? Try to do
Do exponentially distributed random numbers have the memoryless property? Here is one way to find out.Generate many exponentially distributed random numbers with mean 3, using the formula in the previous problem. Find the fraction of them that are greater than
Does the histogram have the shape you would expect?d. Suppose you collected the data in column A by timing arrivals at a store. The value in cell A4 is the time (in minutes) until the first arrival, the value in cell A5 is the time between the first and second arrivals, the value in cell A6 is the
We can easily generate random numbers in a spreadsheet that have an exponential distribution with a given mean. For example, to generate 200 such numbers from an exponential distribution with 13, enter the formula 3*LN(RAND()) in cell A4 and copy it to the range A5:A203. Then select the
Explain the basic relationship between the exponential distribution and a Poisson process. Also, explain how the exponential distribution and the Poisson distribution are fundamentally different. (Hint: What type of data does each describe?)
An extremely important concept in queueing models is the difference between rates and times. If represents a rate (customers per hour, say), then argue why 1 is a time and vice versa.
Austin (1977) conducted an extensive inventory analysis for the United States Air Force. He found that for over 250,000 items the annual holding cost was assumed to equal 32% of the item’s purchase price. He also found that when an order was placed for most items, a fixed cost of over $200 was
Based on Brout (1981). Planner’s Peanuts sells 100 products. The company has been disappointed with the high level of inventory it keeps of each product and its low service level (percentage of demand met on time).Describe how you would help Planner’s improve its performance on both these
A computer manufacturer produces computers for 40 different stores. To monitor its inventory policies, the manufacturer needs to estimate the mean and standard deviation of its weekly demand. How might it do this?
A trucking firm must decide at the beginning of the year on the size of its trucking fleet. If on a given day the firm does not have enough trucks, the firm will have to rent trucks from Hertz. Discuss how you would determine the optimal size of the trucking fleet?
A highly perishable drug spoils after 3 days. A hospital estimates that it is equally likely to need between 1 and 9 units of the drug daily. Each time an order for the drug is placed, a fixed cost of $200 is incurred as well as a purchase cost of $50 per unit. Orders are placed at the end of each
Work the previous problem when the demands are positively correlated, as they might be with products such as peanut butter and jelly. Now use 0.3, 0.5, and 0.7 in your simulations.
The unit cost of each product is$7.50, the unit price for each product is $10, and the unit refund for any unit of either product not sold is$2.50. The company must decide how many units of each product to order. Use @RISK to help the company by experimenting with different order quantities.Try
The correlation between D1 and D2 is , where is a negative number between 1 and
These demands are normally distributed with means 1000 and 1200 and standard deviations 250 and
In most of the Walton Bookstore examples in Chapter 11, we assumed that there is a single product.Suppose instead that a company sells two competing products. Sales of either product tend to take away sales from the other product. That is, the demands for the two products are negatively correlated.
A company currently has two warehouses. Each warehouse services half the company’s demand, and the annual demand serviced by each warehouse is normally distributed with mean 10,000 and standard deviation 1000. The lead time for meeting demand is 1/10 year. The company wants to meet 95% of all
An exchange curve can be used to display the tradeoffs between the average investment in inventory and the annual ordering cost. To illustrate the usefulness of a trade-off curve, suppose that a company must order two products with the attributes shown in the file P13_66.xlsx.a. Draw a curve that
A company inventories two items. The relevant data are shown in the file P13_65.xlsx. Determine the optimal inventory policy if no shortages are allowed and if the average investment in inventory is not allowed to exceed $700. If this constraint could be relaxed by $1, by how much would the
Based on Riccio et al. (1986). The borough of Staten Island has two sanitation districts. In district 1, street litter piles up at an average rate of 2000 tons per week, and in district 2, it piles up at an average rate of 1000 tons per week. Each district has 500 miles of streets.Staten Island has
A firm knows that the price of the product it is ordering is going to increase permanently by X dollars. It wants to know how much of the product it should order before the price increase goes into effect. Here is one approach to this problem. Suppose the firm places one order for Q units before
A newspaper has 500,000 subscribers who pay $4 per month for the paper. It costs the company $200,000 to bill all its customers. Assume that the company can earn interest at a rate of 20% per year on all revenues.Determine how often the newspaper should bill its customers. (Hint: Consider unpaid
The penalty cost p used in the shortage model might be very difficult to estimate. Instead, a company might use a service-level constraint, such as, “95% of all demand must be met from on-hand inventory.” Solve Problem 35 with this constraint instead of the $20,000 penalty cost. Now the problem
Suppose that instead of measuring shortage in terms of cost per shortage per year, a cost of P dollars is incurred for each unit the firm is short. This cost does not depend on the length of time before the backlogged demand is satisfied. Determine a new expression for the annual shortage cost as a
In the previous problem, suppose that the cost per order is $1, and the monthly demand is 50 thermometers.What is the optimal order quantity? What is the smallest discount the supplier could offer that would still be accepted by the hospital?
A hospital orders its thermometers from a hospital supply firm. The cost per thermometer depends on the order quantity Q, as shown in the file P13_58.xlsx.The annual holding cost is 25% of the purchasing cost.Let Q80 be the optimal EOQ order quantity if the cost per thermometer is $0.80, and let
During each year, CSL Computer Company needs to train 27 service representatives. It costs $12,000 to run a training program, regardless of the number of students being trained. Service reps earn a monthly salary of $1500, so CSL does not want to train them before they are needed. Each training
A drugstore sells 30 bottles of antibiotics per week.Each time it orders antibiotics, there is a fixed ordering cost of $10 and a cost of $10 per bottle. Assume that the store’s cost of capital is 10%, there is no storage cost, and antibiotics spoil and cannot be sold if they spend more than 1
In terms of K, D, and h, what is the average length of time that an item spends in inventory before being used to meet demand? Explain how this result can be used to characterize a fast-moving or slow-moving item.
Suppose that instead of ordering the amount Q specified by the EOQ formula, we use the order quantity 0.8Q. Show that the sum of the annual ordering cost and the annual holding cost increases by 2.5%.
Based on Ignall and Kolesar (1972). Father Dominic’s Pizza Parlor receives 30 calls per hour for delivery of pizza. It costs Father Dominic’s $10 to send out a truck to deliver pizzas. Each minute a customer spends waiting for a pizza costs the pizza parlor an estimated$0.20 in lost future
Each time an order is placed, costs of $600 per order and $1500 per computer are incurred. Computers are sold for $2800, and if Computco does not have a computer in stock, the customer will buy a computer from a competitor. At the end of each month, a holding cost of $10 per computer is incurred.
Computco sells personal computers. The demand for its computers during a month follows a normal distribution, with a mean of 400 and standard deviation of
Assume that the cost of holding a TV in inventory for a year is $100. Assume that Lowland begins with 500 TVs in inventory, the cost of a shortage is $150, and the cost of placing an order is $500.a. Suppose that whenever inventory is reviewed, and the inventory level is I, an order for 480I TVs is
Lowland Appliance replenishes its stock of color TVs three times a year. Each order takes 1/9 of a year to arrive. Annual demand for the color TVs follows a normal distribution with a mean of 990 and a standard deviation of
Every 4 years, Blockbuster Publishers revises its textbooks.It has been 3 years since the best-selling book The Joy of Excel has been revised. At present, 2000 copies of the book are in stock, and Blockbuster must determine how many copies of the book to print for the next year. The sales
A department store is trying to decide how many JP Desksquirt II printers to order. Because JP is about to come out with a new model in a few months, the store will order only a limited number of model IIs. The cost per printer is $200, and each printer is sold for$230. If any model IIs are still
A firm experiences demand with a mean of 100 units per day. Lead time demand is normally distributed with mean 1000 units and standard deviation 200 units. It costs $6 to hold 1 unit for 1 year. If the firm wants to meet 90% of all demand on time, what is the expected annual cost of holding safety
A hospital orders its blood from a regional blood bank.Each year, the hospital uses an average of 1040 pints of type O blood. Each order placed with the regional blood bank incurs a cost of $250. The lead time for each order is 5 days. It costs the hospital $20 to hold 1 pint of blood in inventory
How do your answers to part a of the previous problem change if, instead of incurring a $40 penalty cost for each shortage, the store has a service level requirement of meeting 95% of all customer demands on time? In each case (L known with certainty and L random)what penalty cost p is this service
In the previous problem, assume that it costs $300 to place an order. The holding cost per DVD player held in inventory per year is $15. The cost each time a customer orders a DVD player that is not in stock is estimated at $40. (All demand is backlogged.)a. Find the optimal ordering policy for
When the store orders these DVD players from its supplier, it takes an amount of time L for the order to arrive, where L is measured as a fraction of a year. In each of the following, find the mean LD and the standard deviation LD of the demand during lead time.a. Assume that L is known to be
Suppose the annual demand for Soni DVD players at an appliance store is normally distributed with mean 150 and standard deviation
That is, the lead time would then be a certain 3 weeks. What is the most it would be willing to pay (and still meet the service level in part a)?
Chicago’s Treadway Tires Dealer must order tires from its national warehouse. It costs $10,000 to place an order. Annual tire sales are normally distributed with mean 20,000 and standard deviation 5000. It costs $10 per year to hold a tire in inventory, and the lead time for delivery of an order
A hospital must order the drug Porapill from Daisy Drug Company. It costs $500 to place an order. Annual demand for the drug is normally distributed with mean 10,000 and standard deviation 3000, and it costs$5 to hold 1 unit in inventory for 1 year. (A unit is a standard container for the drug.)
A camera store sells an average of 100 cameras per month. The cost of holding a camera in inventory for a year is 30% of the price the camera shop pays for the camera. It costs $120 each time the camera store places an order with its supplier. The price charged per camera depends on the number of
Customers at Joe’s Office Supply Store demand an average of 6000 desks per year. Each time an order is placed, an ordering cost of $300 is incurred. The annual holding cost for a single desk is 25% of the $200 cost of a desk. One week elapses between the placement of an order and the arrival of
It costs $5 to store a unit of either product for a year. The cost of placing an order for either product separately or both products together is $100. Software EG’s annual cost of capital is 14%. Determine a cost-minimizing ordering policy.
The unit purchasing cost is $30 per unit of product 1 and $25 per unit of product
Software EG, a retail company, orders two kinds of software from TeleHard Software. Annually, Software EG sells 800 units of product 1 and 400 units of product
Chicago Mercy Hospital needs to order drugs that are used to treat heart attack victims. Annually, 500 units of drug 1 and 800 units of drug 2 are used. The unit purchasing cost for drug 1 is $150 per unit, and the unit cost of purchasing drug 2 is $300. It costs $20 to store a unit of each drug
The particular logarithmic function proposed in Example 13.4 is just one possibility for the cost of a setup cost reduction. In the previous problem, suppose instead that Machey’s has only three possibilities. The company can either leave the setup cost as it is, spend C1 dollars to reduce the
Reconsider Example 13.1. Each time Machey’s orders cameras, it incurs a $125 ordering cost. Assume that Machey’s could make an investment to decrease this ordering cost. Suppose that any 10% decrease costs a fixed amount, C dollars. Using i 0.10 and Solver, experiment with different values of
A luxury car dealer must pay $20,000 for each car purchased.The annual holding cost is estimated to be 25% of the dollar value of inventory. The dealer sells an average of 500 cars per year. He is willing to backlog some demand but estimates that if he is short one car for one year, he will lose

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