(BQ) Part 2 book An introduction to management science - Quantitative approaches to decision making has contents: Waiting line models, multicriteria decisions, time series analysis and forecasting, markov processes, markov processes, dynamic programming,...and other contents.
CHAPTER 11 Waiting Line Models CONTENTS 11.1 STRUCTURE OF A WAITING LINE SYSTEM Single-Channel Waiting Line Distribution of Arrivals Distribution of Service Times Queue Discipline Steady-State Operation 11.2 SINGLE-CHANNEL WAITING LINE MODEL WITH POISSON ARRIVALS AND EXPONENTIAL SERVICE TIMES Operating Characteristics Operating Characteristics for the Burger Dome Problem Managers’ Use of Waiting Line Models Improving the Waiting Line Operation Excel Solution of Waiting Line Model 11.3 MULTIPLE-CHANNEL WAITING LINE MODEL WITH POISSON ARRIVALS AND EXPONENTIAL SERVICE TIMES Operating Characteristics Operating Characteristics for the Burger Dome Problem 11.4 SOME GENERAL RELATIONSHIPS FOR WAITING LINE MODELS 11.5 ECONOMIC ANALYSIS OF WAITING LINES 11.6 OTHER WAITING LINE MODELS 11.7 SINGLE-CHANNEL WAITING LINE MODEL WITH POISSON ARRIVALS AND ARBITRARY SERVICE TIMES Operating Characteristics for the M/G/1 Model Constant Service Times 11.8 MULTIPLE-CHANNEL MODEL WITH POISSON ARRIVALS, ARBITRARY SERVICE TIMES, AND NO WAITING LINE Operating Characteristics for the M/G/k Model with Blocked Customers Cleared 11.9 WAITING LINE MODELS WITH FINITE CALLING POPULATIONS Operating Characteristics for the M/M/1 Model with a Finite Calling Population Chapter 11 Waiting Line Models 503 Recall the last time that you had to wait at a supermarket checkout counter, for a teller at your local bank, or to be served at a fast-food restaurant In these and many other waiting line situations, the time spent waiting is undesirable Adding more checkout clerks, bank tellers, or servers is not always the most economical strategy for improving service, so businesses need to identify other ways to keep waiting times within tolerable limits Models have been developed to help managers understand and make better decisions concerning the operation of waiting lines In management science terminology, a waiting line is also known as a queue, and the body of knowledge dealing with waiting lines is known as queueing theory In the early 1900s, A K Erlang, a Danish telephone engineer, began a study of the congestion and waiting times occurring in the completion of telephone calls Since then, queueing theory has grown far more sophisticated, with applications in a wide variety of waiting line situations Waiting line models consist of mathematical formulas and relationships that can be used to determine the operating characteristics (performance measures) for a waiting line Operating characteristics of interest include the following: The probability that no units are in the system The average number of units in the waiting line The average number of units in the system (the number of units in the waiting line plus the number of units being served) The average time a unit spends in the waiting line The average time a unit spends in the system (the waiting time plus the service time) The probability that an arriving unit has to wait for service Managers who have such information are better able to make decisions that balance desirable service levels against the cost of providing the service The Management Science in Action, ATM Waiting Times at Citibank, describes how a waiting line model was used to help determine the number of automatic teller machines (ATMs) to place at New York City banking centers A waiting line model prompted the creation of a new kind of line and a chief line director to implement first-come, first-served queue discipline at Whole Foods Market in the Chelsea neighborhood of New York City In addition, a waiting line model helped the New Haven, Connecticut, fire department develop policies to improve response time for both fire and medical emergencies MANAGEMENT SCIENCE IN ACTION ATM WAITING TIMES AT CITIBANK* The waiting line model used at Citibank is discussed in Section 11.3 The New York City franchise of U.S Citibanking operates more than 250 banking centers Each center provides one or more automatic teller machines (ATMs) capable of performing a variety of banking transactions At each center, a waiting line is formed by randomly arriving customers who seek service at one of the ATMs In order to make decisions on the number of ATMs to have at selected banking center locations, management needed information about potential waiting times and general customer service Waiting line operating characteristics such as average number of customers in the waiting line, average time a customer spends waiting, and the probability that an arriving customer has to wait would help management determine the number of ATMs to recommend at each banking center For example, one busy midtown Manhattan center had a peak arrival rate of 172 customers per hour A multiple-channel waiting line model with six ATMs showed that 88% of the customers would have to wait, with an average wait time (continued) 504 Chapter 11 Waiting Line Models between six and seven minutes This level of service was judged unacceptable Expansion to seven ATMs was recommended for this location based on the waiting line model’s projection of acceptable waiting times Use of the waiting line model 11.1 provided guidelines for making incremental ATM decisions at each banking center location *Based on information provided by Stacey Karter of Citibank STRUCTURE OF A WAITING LINE SYSTEM To illustrate the basic features of a waiting line model, we consider the waiting line at the Burger Dome fast-food restaurant Burger Dome sells hamburgers, cheeseburgers, french fries, soft drinks, and milk shakes, as well as a limited number of specialty items and dessert selections Although Burger Dome would like to serve each customer immediately, at times more customers arrive than can be handled by the Burger Dome food service staff Thus, customers wait in line to place and receive their orders Burger Dome is concerned that the methods currently used to serve customers are resulting in excessive waiting times Management wants to conduct a waiting line study to help determine the best approach to reduce waiting times and improve service Single-Channel Waiting Line In the current Burger Dome operation, a server takes a customer’s order, determines the total cost of the order, takes the money from the customer, and then fills the order Once the first customer’s order is filled, the server takes the order of the next customer waiting for service This operation is an example of a single-channel waiting line Each customer entering the Burger Dome restaurant must pass through the one channel—one order-taking and order-filling station—to place an order, pay the bill, and receive the food When more customers arrive than can be served immediately, they form a waiting line and wait for the order-taking and order-filling station to become available A diagram of the Burger Dome single-channel waiting line is shown in Figure 11.1 Distribution of Arrivals Defining the arrival process for a waiting line involves determining the probability distribution for the number of arrivals in a given period of time For many waiting line situations, the arrivals occur randomly and independently of other arrivals, and we cannot predict FIGURE 11.1 THE BURGER DOME SINGLE-CHANNEL WAITING LINE System Server Customer Arrivals Waiting Line Order Taking and Order Filling Customer Leaves after Order Is Filled 11.1 505 Structure of a Waiting Line System when an arrival will occur In such cases, quantitative analysts have found that the Poisson probability distribution provides a good description of the arrival pattern The Poisson probability function provides the probability of x arrivals in a specific time period The probability function is as follows:1 P(x) = lxe-l x! for x = 0, 1, 2, (11.1) where x = the number of arrivals in the time period l = the mean number of arrivals per time period e = 2.71828 The mean number of arrivals per time period, l, is called the arrival rate Values of eϪl can be found with a calculator or by using Appendix C Suppose that Burger Dome analyzed data on customer arrivals and concluded that the arrival rate is 45 customers per hour For a one-minute period, the arrival rate would be l ϭ 45 customers͞60 minutes ϭ 0.75 customers per minute Thus, we can use the following Poisson probability function to compute the probability of x customer arrivals during a one-minute period: P(x) = lxe-l 0.75xe-0.75 = x! x! (11.2) Thus, the probabilities of 0, 1, and customer arrivals during a one-minute period are P(0) = P(1) = P(2) = (0.75)0e-0.75 0! (0.75)1e-0.75 1! (0.75)2e-0.75 2! = e-0.75 = 0.4724 = 0.75e-0.75 = 0.75(0.4724) = 0.3543 (0.75)2e-0.75 = 2! (0.5625)(0.4724) = = 0.1329 The probability of no customers in a one-minute period is 0.4724, the probability of one customer in a one-minute period is 0.3543, and the probability of two customers in a oneminute period is 0.1329 Table 11.1 shows the Poisson probabilities for customer arrivals during a one-minute period The waiting line models that will be presented in Sections 11.2 and 11.3 use the Poisson probability distribution to describe the customer arrivals at Burger Dome In practice, you should record the actual number of arrivals per time period for several days or weeks and compare the frequency distribution of the observed number of arrivals to the Poisson probability distribution to determine whether the Poisson probability distribution provides a reasonable approximation of the arrival distribution The term x!, x factorial, is defined as x! ϭ x(x Ϫ 1)(x Ϫ 2) (2)(1) For example, 4! ϭ (4)(3)(2)(1) ϭ 24 For the special case of x ϭ 0, 0! ϭ by definition 506 Chapter 11 Waiting Line Models TABLE 11.1 POISSON PROBABILITIES FOR THE NUMBER OF CUSTOMER ARRIVALS AT A BURGER DOME RESTAURANT DURING A ONE-MINUTE PERIOD (l ϭ 0.75) Number of Arrivals or more Probability 0.4724 0.3543 0.1329 0.0332 0.0062 0.0010 Distribution of Service Times The service time is the time a customer spends at the service facility once the service has started At Burger Dome, the service time starts when a customer begins to place the order with the food server and continues until the customer receives the order Service times are rarely constant At Burger Dome, the number of items ordered and the mix of items ordered vary considerably from one customer to the next Small orders can be handled in a matter of seconds, but large orders may require more than two minutes Quantitative analysts have found that if the probability distribution for the service time can be assumed to follow an exponential probability distribution, formulas are available for providing useful information about the operation of the waiting line Using an exponential probability distribution, the probability that the service time will be less than or equal to a time of length t is P(service time … t) = - e-mt (11.3) where m = the mean number of units that can be served per time period e = 2.71828 A property of the exponential probability distribution is that there is a 0.6321 probability that the random variable takes on a value less than its mean In waiting line applications, the exponential probability distribution indicates that approximately 63 percent of the service times are less than the mean service time and approximately 37 percent of the service times are greater than the mean service time The mean number of units that can be served per time period, , is called the service rate Suppose that Burger Dome studied the order-taking and order-filling process and found that the single food server can process an average of 60 customer orders per hour On a oneminute basis, the service rate would be ϭ 60 customers͞60 minutes ϭ customer per minute For example, with m ϭ 1, we can use equation (11.3) to compute probabilities such as the probability an order can be processed in ¹⁄₂ minute or less, minute or less, and minutes or less These computations are P(service time … 0.5 min.) = - e-1(0.5) = - 0.6065 = 0.3935 P(service time … 1.0 min.) = - e-1(1.0) = - 0.3679 = 0.6321 P(service time … 2.0 min.) = - e-1(2.0) = - 0.1353 = 0.8647 Thus, we would conclude that there is a 0.3935 probability that an order can be processed in ¹⁄₂ minute or less, a 0.6321 probability that it can be processed in minute or less, and a 0.8647 probability that it can be processed in minutes or less 11.1 Structure of a Waiting Line System 507 In several waiting line models presented in this chapter, we assume that the probability distribution for the service time follows an exponential probability distribution In practice, you should collect data on actual service times to determine whether the exponential probability distribution is a reasonable approximation of the service times for your application Queue Discipline In describing a waiting line system, we must define the manner in which the waiting units are arranged for service For the Burger Dome waiting line, and in general for most customer-oriented waiting lines, the units waiting for service are arranged on a first-come, first-served basis; this approach is referred to as an FCFS queue discipline However, some situations call for different queue disciplines For example, when people wait for an elevator, the last one on the elevator is often the first one to complete service (i.e., the first to leave the elevator) Other types of queue disciplines assign priorities to the waiting units and then serve the unit with the highest priority first In this chapter we consider only waiting lines based on a first-come, first-served queue discipline The Management Science in Action, The Serpentine Line and an FCFS Queue Discipline at Whole Foods Market, describes how an FCFS queue discipline is used at a supermarket MANAGEMENT SCIENCE IN ACTION THE SERPENTINE LINE AND AN FCFS QUEUE DISCIPLINE AT WHOLE FOODS MARKET* The Whole Foods Market in the Chelsea neighborhood of New York City employs a chief line director to implement a first-come, first-served (FCFS) queue discipline Companies such as Wendy’s, American Airlines, and Chemical Bank were among the first to employ serpentine lines to implement an FCFS queue discipline Such lines are commonplace today We see them at banks, amusement parks, and fastfood outlets The line is called serpentine because of the way it winds around When a customer gets to the front of the line, the customer then goes to the first available server People like serpentine lines because they prevent people who join the line later from being served ahead of an earlier arrival As popular as serpentine lines have become, supermarkets have not employed them because of a lack of space At the typical supermarket, a separate line forms at each checkout counter When ready to check out, a person picks one of the checkout counters and stays in that line until receiving service Sometimes a person joining another checkout line later will receive service first, which tends to upset people Manhattan’s Whole Foods Market solved this problem by creating a new kind of line and employing a chief line director to direct the first person in line to the next available checkout counter The waiting line at the Whole Foods Market is actually three parallel lines Customers join the shortest line and follow a rotation when they reach the front of the line For instance, if the first customer in line is sent to a checkout counter, the next customer sent to a checkout counter is the first person in line 2, then the first person in line 3, and so on This way an FCFS queue discipline is implemented without a long, winding serpentine line The Whole Foods Market’s customers seem to really like the system, and the line director, Bill Jones, has become something of a celebrity Children point to him on the street and customers invite him over for dinner *Based on Ian Parker, “Mr Next,” The New Yorker (January 13, 2003) Steady-State Operation When the Burger Dome restaurant opens in the morning, no customers are in the restaurant Gradually, activity builds up to a normal or steady state The beginning or start-up period is referred to as the transient period The transient period ends when the system reaches 508 Chapter 11 Waiting Line Models the normal or steady-state operation Waiting line models describe the steady-state operating characteristics of a waiting line 11.2 Waiting line models are often based on assumptions such as Poisson arrivals and exponential service times When applying any waiting line model, data should be collected on the actual system to ensure that the assumptions of the model are reasonable SINGLE-CHANNEL WAITING LINE MODEL WITH POISSON ARRIVALS AND EXPONENTIAL SERVICE TIMES In this section we present formulas that can be used to determine the steady-state operating characteristics for a single-channel waiting line The formulas are applicable if the arrivals follow a Poisson probability distribution and the service times follow an exponential probability distribution As these assumptions apply to the Burger Dome waiting line problem introduced in Section 11.1, we show how formulas can be used to determine Burger Dome’s operating characteristics and thus provide management with helpful decisionmaking information The mathematical methodology used to derive the formulas for the operating characteristics of waiting lines is rather complex However, our purpose in this chapter is not to provide the theoretical development of waiting line models, but rather to show how the formulas that have been developed can provide information about operating characteristics of the waiting line Readers interested in the mathematical development of the formulas can consult the specialized texts listed in Appendix D at the end of the text Operating Characteristics The following formulas can be used to compute the steady-state operating characteristics for a single-channel waiting line with Poisson arrivals and exponential service times, where l = the mean number of arrivals per time period (the arrival rate) m = the mean number of services per time period (the service rate) The probability that no units are in the system: Equations (11.4) through (11.10) not provide formulas for optimal conditions Rather, these equations provide information about the steady-state operating characteristics of a waiting line P0 = - l m (11.4) The average number of units in the waiting line: Lq = l2 m(m - l) (11.5) The average number of units in the system: L = Lq + l m (11.6) The average time a unit spends in the waiting line: Wq = Lq l (11.7) 11.2 Single-Channel Waiting Line Model with Poisson Arrivals and Exponential Service Times 509 The average time a unit spends in the system: W = Wq + m (11.8) The probability that an arriving unit has to wait for service: Pw = l m (11.9) The probability of n units in the system: l n Pn = a b P0 m (11.10) The values of the arrival rate l and the service rate µ are clearly important components in determining the operating characteristics Equation (11.9) shows that the ratio of the arrival rate to the service rate, l͞m, provides the probability that an arriving unit has to wait because the service facility is in use Hence, l͞m is referred to as the utilization factor for the service facility The operating characteristics presented in equations (11.4) through (11.10) are applicable only when the service rate m is greater than the arrival rate l—in other words, when l͞m Ͻ If this condition does not exist, the waiting line will continue to grow without limit because the service facility does not have sufficient capacity to handle the arriving units Thus, in using equations (11.4) through (11.10), we must have m Ͼ l Operating Characteristics for the Burger Dome Problem Recall that for the Burger Dome problem we had an arrival rate of l ϭ 0.75 customers per minute and a service rate of m ϭ customer per minute Thus, with m Ͼ l, equations (11.4) through (11.10) can be used to provide operating characteristics for the Burger Dome single-channel waiting line: l 0.75 = = 0.25 m l2 0.752 = = 2.25 customers m(m - l) 1(1 - 0.75) l 0.75 Lq + = customers = 2.25 + m Lq 2.25 = = minutes l 0.75 1 = + = minutes Wq + m l 0.75 = = 0.75 m P0 = Lq = L = Wq = W = Pw = 510 Chapter 11 Waiting Line Models TABLE 11.2 THE PROBABILITY OF n CUSTOMERS IN THE SYSTEM FOR THE BURGER DOME WAITING LINE PROBLEM Number of Customers or more Problem asks you to compute the operating characteristics for a singlechannel waiting line application Probability 0.2500 0.1875 0.1406 0.1055 0.0791 0.0593 0.0445 0.1335 Equation (11.10) can be used to determine the probability of any number of customers in the system Applying it provides the probability information in Table 11.2 Managers’ Use of Waiting Line Models The results of the single-channel waiting line for Burger Dome show several important things about the operation of the waiting line In particular, customers wait an average of three minutes before beginning to place an order, which appears somewhat long for a business based on fast service In addition, the facts that the average number of customers waiting in line is 2.25 and that 75% of the arriving customers have to wait for service are indicators that something should be done to improve the waiting line operation Table 11.2 shows a 0.1335 probability that seven or more customers are in the Burger Dome system at one time This condition indicates a fairly high probability that Burger Dome will experience some long waiting lines if it continues to use the single-channel operation If the operating characteristics are unsatisfactory in terms of meeting company standards for service, Burger Dome’s management should consider alternative designs or plans for improving the waiting line operation Improving the Waiting Line Operation Waiting line models often indicate when improvements in operating characteristics are desirable However, the decision of how to modify the waiting line configuration to improve the operating characteristics must be based on the insights and creativity of the analyst After reviewing the operating characteristics provided by the waiting line model, Burger Dome’s management concluded that improvements designed to reduce waiting times are desirable To make improvements in the waiting line operation, analysts often focus on ways to improve the service rate Generally, service rate improvements are obtained by making either or both of the following changes: Increase the service rate by making a creative design change or by using new technology Add one or more service channels so that more customers can be served simultaneously Assume that in considering alternative 1, Burger Dome’s management decides to employ an order filler who will assist the order taker at the cash register The customer begins the service process by placing the order with the order taker As the order is placed, the order taker announces the order over an intercom system, and the order filler begins filling the 11.2 Single-Channel Waiting Line Model with Poisson Arrivals and Exponential Service Times 511 TABLE 11.3 OPERATING CHARACTERISTICS FOR THE BURGER DOME SYSTEM WITH THE SERVICE RATE INCREASED TO μ ϭ 1.25 CUSTOMERS PER MINUTE Probability of no customers in the system Average number of customers in the waiting line Average number of customers in the system Average time in the waiting line Average time in the system Probability that an arriving customer has to wait Probability that seven or more customers are in the system Problem 11 asks you to determine whether a change in the service rate will meet the company’s service guideline for its customers 0.400 0.900 1.500 1.200 minutes 2.000 minutes 0.600 0.028 order When the order is completed, the order taker handles the money, while the order filler continues to fill the order With this design, Burger Dome’s management estimates the service rate can be increased from the current 60 customers per hour to 75 customers per hour Thus, the service rate for the revised system is m ϭ 75 customers͞60 minutes ϭ 1.25 customers per minute For l ϭ 0.75 customers per minute and m ϭ 1.25 customers per minute, equations (11.4) through (11.10) can be used to provide the new operating characteristics for the Burger Dome waiting line These operating characteristics are summarized in Table 11.3 The information in Table 11.3 indicates that all operating characteristics have improved because of the increased service rate In particular, the average time a customer spends in the waiting line has been reduced from to 1.2 minutes and the average time a customer spends in the system has been reduced from to minutes Are any other alternatives available that Burger Dome can use to increase the service rate? If so, and if the mean service rate m can be identified for each alternative, equations (11.4) through (11.10) can be used to determine the revised operating characteristics and any improvements in the waiting line system The added cost of any proposed change can be compared to the corresponding service improvements to help the manager determine whether the proposed service improvements are worthwhile As mentioned previously, another option often available is to add one or more service channels so that more customers can be served simultaneously The extension of the singlechannel waiting line model to the multiple-channel waiting line model is the topic of the next section Excel Solution of Waiting Line Model Waiting line models are easily implemented with the aid of worksheets The Excel worksheet for the Burger Dome single-channel waiting line is shown in Figure 11.2 The formula worksheet is in the background; the value worksheet is in the foreground The arrival rate and the service rate are entered in cells B7 and B8 The formulas for the waiting line’s operating characteristics are placed in cells C13 to C18 The worksheet shows the same values for the operating characteristics that we obtained earlier Modifications in the waiting line design can be evaluated by entering different arrival rates and/or service rates into cells B7 and B8 The new operating characteristics of the waiting line will be shown immediately The Excel worksheet in Figure 11.2 is a template that can be used with any singlechannel waiting line model with Poisson arrivals and exponential service times This worksheet and similar Excel worksheets for the other waiting line models presented in this chapter are available at the website that accompanies this text Appendix E Chapter 16 N = (I - Q)-1 = c a 0.82 b p1 ϭ 0.5, p2 ϭ 0.5 c p1 ϭ 0.6, p2 ϭ 0.4 NR = c a 0.10 as given by the transition probability (1) b p1 ϭ 0.90p1 ϩ 0.30p2 p2 ϭ 0.10p1 ϩ 0.70p2 (2) p1 ϩ p2 ϭ (1) Using (1) and (3), 0.10p1 Ϫ 0.30p2 ϭ 0.10 p1 Ϫ 0.30 (1 Ϫ p1) ϭ 0.10 p1 Ϫ 0.30 ϩ 0.30p1 ϭ 0.40p1 ϭ 0.30 p1 ϭ 0.75 p2 ϭ (1 Ϫ p1) ϭ 0.25 a p1 ϭ 0.92, p2 ϭ 0.08 b $85 City Suburbs Suburbs 0.02 0.99 b p1 ϭ 0.333, p2 ϭ 0.667 c City will decrease from 40% to 33%; suburbs will increase from 60% to 67% a p1 ϭ 0.85p1 ϩ 0.20p2 ϭ 0.15p3 (1) p2 ϭ 0.10p1 ϩ 0.75p2 ϭ 0.10p3 (2) p3 ϭ 0.05p1 ϩ 0.05p2 ϭ 0.75p3 (3) p1 ϩ p2 ϩ p3 ϭ (4) Using (1), (2), and (4) provides three equations with three unknowns; solving provides p1 ϭ 0.548, p2 ϭ 0.286, and p3 ϭ 0.166 b 16.6% as given by p3 c Quick Stop should take 667 Ϫ 0.548(1000) ϭ 119 Murphy’s customers and 333 Ϫ 0.286(1000)ϭ 47 Ashley’s customers Total 166 Quick Stop customers It will take customers from Murphy’s and Ashley’s a MDA b p1 ϭ 1⁄ 3, p2 ϭ 2⁄ 10 Ϫ 1(0.59), Ϫ 1(0.52) 11 I = c d (I - Q) = c Q = c 0.75 -0.05 0.25 0.05 -0.25 d 0.75 0.25 d 0.25 1.3636 0.4545 d 0.0909 1.3636 1.3636 0.4545 0.5 dc 0.0909 1.3636 0.5 0.0 0.909 d = c 0.2 0.727 0.909 0.091 d = [7271 0.727 0.273 Estimate $1729 in bad debts BNR = [4000 5000]c 12 3580 will be sold eventually; 1420 will be lost 14 a Graduate and Drop Out b P(Drop Out) ϭ 0.15, P(Sophomore) ϭ 0.10, P(Junior) ϭ 0.75 c 0.706, 0.294 d Yes; P(Graduate) ϭ 0.54 P(Drop Out) ϭ 0.46 e 1479 (74%) will graduate Appendix A a City 0.98 0.01 851 Self-Test Solutions and Answers to Even-Numbered Problems ϭF6*$F$3 0.091 d 0.273 1729] 852 Appendix E Self-Test Solutions and Answers to Even-Numbered Problems Cell Formula D14 E14 F14 G14 H14 I14 ϭC14*$B$3 ϭC14*$B$7 ϭC14*$B$9 ϭ$B$5 ϭSUM(E14:G14) ϭD14-H14 10 Error in SUMPRODUCT range in cell B17 Cell A23 should be Lexington Index “Note: Entries accompanied by n indicate notes Chapters 17 through 21 are found on the accompanying website and are indicated by the chapter number followed by the page number (i.e., 17-5)” A Absorbing state probabilities, 781–782 Accounts receivable analysis, 771–775 Accuracy, of forecasts, 713–717 exponential smoothing and, 723–725 moving averages for, 718–720 Activity times, in project scheduling, 413 crashing, 432–433 scheduling with known times for, 413–422 scheduling with unknown times for, 422–430 time-cost tradeoffs for, 431 Additivity, 34n Advertising advertising campaign planning, 202–203 media selection application of linear programming for, 155–158 Airline industry Air New Zealand, 318–319 American Airlines, 2–3, 223, 224 Bombardier Flexjet, 366–367 preboard airport screening simulation, 573 revenue management used by, 223–230 simulation of overbooking in, 543 waiting line problems for reservations in, 539–540 Air New Zealand, 318–319 All-integer linear programs, 319–321 computer solutions of, 324–325 graphical solutions of, 322–324 Alternative optimal solutions, 57–58 American Airlines, 2–3 revenue management used by, 223, 224 Analog models, Analytic hierarchy process (AHP), 17, 679–680 consistency in, 685–686 pairwise comparisons in, 681–683, 687–688 priority rankings developed in, 688–689 software for, 689n synthesization in, 684–685 Annuities, variable, 236 Arcs, 19-8–19-9 in networks, 257 in project networks, 414 Arrival rates, 505 Arrival times, 564–565 Artificial variables, 17-21 Asset allocation, 236 Assignment problems, 263–268, 19-2 Excel for, 311–313 Hungarian method, 19-18–19-21 ASW Publishing, Inc., 357–358 AT&T, 283 @Risk (software), 557, 574 Automatic teller machines (ATMs), 503–504 simulation of, 563–574, 594–597 Automobile industry car rental industry, 224, 479–480 environmental compliance in, 374, 405–407 Ford Motor Company, 678 Porsche Shop, 443–444 Averages See Means B Backorders, 468, 471, 472n Backward passes, in project networks, 417 Banking applications, 215–216 automatic teller machines, 503–504 bank location problems, 334–337 simulation of automatic teller machines, 563–574, 594–597 Basic feasible solutions, 17-4–17-5 Basic solutions, 17-4 Basic variables, 17-4, 17-9 Bayer Pharmaceuticals, 629 Bayes' theorem, 630–633 Bellman, Richard, 21-2 Best-case scenarios, 546 Beta probability distributions, 425 Binary expansion, 340n Binary variables, 318 Blackjack (twenty-one) absorbing state probabilities in, 781–782 simulation of, 581 Blending problems, 183–188, 207–209 nonlinear optimization for, 382–387 Bombardier Flexjet (firm), 366–367 Branches adding, in TreePlan, 655 computing branch probabilities, 630–634 in decision trees, 607 Breakeven analysis, 16 Excel for, 24–27 C Call centers, 544 Calling populations, finite and infinite, 526–529 Canonical form, 18-14 854 Index Capacitated transportation problems, 263 Capital budgeting problems, 325–329 Car rental industry, 224, 479–480 Central limit theorem, 429n Chance events, 604 Chance nodes, 605 adding, in TreePlan, 655–656 Citibank, 503–504 Clean Energy Act (U.S., 2007), 374 Communications networks, 20-5 Computer industry, call centers in, 544 Computers for all-integer linear programs, 324–325 decision analysis software for, 616, 620n for goal programming, 671–673 linear programming applications of, 50–52, 109n sensitivity analysis applications of, 103–110 Computer simulations implementation of, 574–575 random numbers generated for, 549–551 of sensitivity analysis problem, 118–122 Concave functions, 371 Conditional constraints, 342 Conditional probabilities, 632 Consequence nodes, 605 Conservative approach, in decision making, 607–608 Consistency, in analytic hierarchy process, 685–686 Consistency ratios, 685–686 Constant demand rate assumption, 454–455 Constant service times, 523 Constant supply rate assumption, 464 Constrained nonlinear optimization problems, 367–371 Constraint coefficients, 112 Constraints conditional and corequisite, 342 in goal programming, 661–663, 667n, 668n multiple-choice and mutually exclusive, 341–342 redundant constraints, 48 Continuous review inventory systems, 484 Contour lines, 370–371 Controllable inputs, 543 Convery, John, 21-21 Convex functions, 372 Corequisite constraints, 342 Corporate Average Fuel Economy (CAFE) regulations, 374, 405–407 Costs backorder costs, 471 fixed, in capital budgeting problems, 326–329 goodwill costs, 468 holding costs, 456–458 models of, 14–15 in simulations, 547 time-cost tradeoffs, in project scheduling, 431–436 in total cost models, 455n, 465–467 of waiting line channels, 519–520 Crashing, in project scheduling, 431 of activity times, 432–433 linear programming model for, 434–436 Crew scheduling problems, 318–319, 340–341 Critical activities, 416, 420 Critical Path Method (CPM), 17, 413 for project scheduling with known activity times, 413–422 Critical paths, in project scheduling with known activity times, 414–420 Microsoft Office Project for, 450–452 with unknown activity times, 425–428 Crystal Ball (software), 557n, 574, 597–601 Curling (sport), 776 Curve fitting models, 724–725 Excel Solver for, 758–759 for exponential trend equation, 733 for liner trends, 726–729 for seasonality, 737 software for, 730n Customer arrival times, 564–565 Customer service times, 565 CVS Corporation, 454 Cycle time, 461 Cyclical patterns, 713 D Dantzig, George, 2, 30, 17-3 Data, preparation of, for models, 10–11 Data envelopment analysis (DEA), 215–223 banking applications of, 215–216 Decision alternatives, 604 Decision analysis, 17, 603–604 branch probabilities for, 630–634 at Eastman Kodak, 603 with probabilities, 610–615 problem formulation in, 604–610 risk analysis in, 615–616 with sample information, 620–630 sensitivity analysis in, 616–620 TreePlan for, 653–658 without probabilities, 607–610 Decision making multicriteria decisions, 660 problem solving and, 3–4 quantitative analysis in, 4–6 Decision nodes, 605 Decision strategies, 623–627 Decision trees, 606–607, 621–623 TreePlan for, 653–658 Decision variables, in blending problems, 188n dn, 21-7 in transportation problems, 258, 262n Decomposition methods, 366–367 Deere & Company, 489 Definitional variables, 234n Degeneracy, 110n, 17-29, 17-33–17-35 855 Index Degenerate solutions, 19-12 Delta Airlines, 17-2 Demand See also Inventory models constant demand rate assumption for, 454–455 in order-quantity, reorder point inventory model, with probabilistic demand, 480–484 in periodic review inventory model with probabilistic demand, 484–488 quantity discounts for, 472–474 shortages and, 467–472 in simulation model, 557 single-period inventory models with probabilistic demand, 474–480 in transportation problems, 260 Dennis, Greg A., 20-5 Deterministic models, 9–10 inventory models, 474 DIRECTV (firm), 391–392 Disability claims, 612–613 Discounts, quantity discounts, 472–474 Discrete-event simulation models, 563, 573–574n Distribution models, 17 for assignment problems, 263–268 at Kellogg Company, 180 maximal flow problems, 279–283 for shortest-route problems, 276–279 for transportation problems, 256–263 for transshipment problem, 268–275 Distribution system design problems, 329–334 “Divide and conquer” solution strategy, 21-10 Divisibility, 34n Drive-through waiting line simulations, 589–590 Drug decision analysis, 629 Drugstore industry, CVS Corporation, 454 Duality, 18-14–18-20 Dual prices, 18-6, 18-18 absolute value of, 18-12n Dual problems, 18-14 Dual values, 101–102 caution regarding, 106 in computer solution, 104–105 nonintuitive, 112–114 in nonlinear optimization problems, 374 Dual variables, 18-14, 18-16–18-18 Duke Energy Corporation, 207–209, 704–705 Duke University, 634 Dummy agents, in assignment problems, 267n Dummy destinations, 19–18 Dummy origins, 19-2, 19-17, 19-19 in transportation problems, 260 Dummy variables in monthly data, 739–740 for seasonality without trends, 734–735 for seasonality with trends, 737, 739 Dynamic programming knapsack problem, 21-10–21-16 notation for, 21-6–21-10 production and inventory control problem, 21-16–21-20 shortest-route problem, 21-2–21-6 stages of, 21-6 Dynamic simulation models, 563 E Eastman Kodak, 93, 603 Economic order quantity (EOQ) formula, 459 Economic order quantity (EOQ) model, 454–459 Excel for, 462–463 optimal order quantity formula for, 500–501 order-quantity, reorder point model with probabilistic demand, 480–484 order quantity decision in, 459–460 quantity discounts in, 472–474 sensitivity analysis for, 461–462 time of order decision in, 460–461 Economic production lot size model, 464–467 optimal lot size formula for, 501 Edmonton Folk Festival, 340–341 EDS, 20-5 Efficiency data envelopment analysis to identify, 215–223 of sample information, 630 warehouse efficiency simulation, 576–577 Eisner, Mark, 18 Electricity generation, 704–705 Elementary row operations, 17-12 Energy industry, 704–705 Environmental Protection Agency (EPA, U.S.), 21-21 Environmental regulation, in automobile industry, 374, 405–407 Erlang, A K., 503 Events, in simulations, 563 Excel, 14n for assignment problems, 311–313 for breakeven analysis, 24–27 for economic order quantity model, 462–463 for financial planning problem, 210–213 for forecasting, 753–754 to generate random numbers, 549, 552 for integer linear programming, 360–364 for inventory simulation, 561–562 for linear programming, 87–91 for matrix inversions, 785–786 for nonlinear optimization problems, 409–411 for scoring models, 701–702 for sensitivity analysis, 148–150 for simulations, 554–556, 590–597 for transportation problems, 308–310 for transshipment problems, 313–315 TreePlan add-in for, 653–658 for waiting line models, 511–512 for waiting line simulation, 569–573 Excel Solver for curve fitting, 754–759 for integer linear programming, 361 856 Index Excel Solver (continued) for linear programming, 89 for nonlinear optimization problems, 409–411 for sensitivity analysis, 148 for transshipment problem, 314 Expected times, in project scheduling, 425 Expected value approach, 610–612 Expected values (EVs), 610 of perfect information, 613–615 of sample information, 629–630 sensitivity analysis for, 617–620 Expert Choice (software), 689n Exponential probability distributions, 506 in multiple-channel waiting line model, with Poisson arrivals, 512–517 in single-channel waiting line model, with Poisson arrivals, 508–512 Exponential smoothing, 721–725 Excel for, 753–754 Excel Solver for, 754–758 spreadsheets for, 726n Exponential trend equation, 733 Extreme points, in linear programming, 48–50 F False-positive and false-negative test results, 634 Fannon, Vicki, 341 Feasible region, 38 Feasible solutions, 38–42 infeasibility and, 58–60 Financial applications of linear programming, 161 banking applications, 215–216 capital budgeting problems, 325–329 financial planning applications of, 164–172, 210–213 portfolio models, 229–235 portfolio selection, 161–164 revenue management applications, 223–229 Financial planning applications, 164–172 Excel for, 210–213 index fund construction as, 374–379 Markowitz portfolio model for, 379–382 simulation of, 585–587 Finite calling populations, 526–529 First-come, first served (FCFS) waiting lines, 507 Fixed costs, 14 in capital budgeting problems, 326–329 Fleet assignment, 17-2 Flow capacity, 279 Little's flow equations for, 517–518 simulation in, 544 Flow problems See Network flow problems Ford Motor Company, 678 Forecast errors, 713–716 Forecasting, 18, 704 accuracy of, 713–717 adoption of new products, 387–391 Excel for, 753–754 Excel Solver for, 754–759 LINGO for, 759–760 methods used for, 713 moving averages for, 717–720 in utility industry, 704–705 Forward passes, in project networks, 417 Four-stage dynamic programming procedure, 21-6 Free variables, 378n Fundamental matrices, 772–774 G Game theory, 236–247 Gantt, Henry L., 413 Gantt Charts, 413 Microsoft Office Project for, 451 General Accountability Office (GAO, U.S.), 762 General Electric Capital, 164–165 General Electric Plastics (GEP), 110 Global optima, 371–374 Goal constraints, 667n Goal equations, 661–663 for multiple goals at same priority level, 669–670 Goal programming, 17, 660–663 complex problems in, 668–674 graphical solution procedures for, 664–667 model used in, 667 objective functions with preemptive priorities for, 663 scoring models for, 674–678 Golf course tee time reservation simulation, 587–589 Goodwill costs, 468 Graphical solution procedures, 35–48 for all-integer linear programs, 322–324 for goal programming, 664–667 for minimization, 54–55 in sensitivity analysis, 95–103 Greedy algorithms, 20-5n H Harmonic averages, 405, 406 Harris, F W., 459 Heery International, 268 Heuristics, 19-2 Hewlett-Packard, 454 Hierarchies, in analytic hierarchy process, 679–680 Holding costs, 456–458, 464 Horizontal patterns, 705–707 Hospitals bonds issued for financing of, 422 performance of, 216–222 Human resources training versus hiring decisions, 203–204 workforce assignment applications in, 179–183, 205–207 Hungarian method, 19-18–19-21 Hypothetical composites, 216–217 857 Index I IBM Corporation, 454 Iconic models, Immediate predecessors, in project scheduling, 413 Incoming arcs, 19-19 Incremental analysis, 476 Index funds, 374–379 Infeasibility, 58–60, 61n, 17-29–17-31, 17-35 goal programming for, 674n Infinite calling populations, 526 Influence diagrams, 605, 620–621 Inputs, to models, random, generating, 549–554 in simulations, 543 Insurance industry, portfolio models used in, 236 Integer linear programming, 16, 318 all-integer linear programs for, 321–324 for bank location problems, 334–337 for crew scheduling problem, 318–319 for distribution system design problems, 329–334 Excel, 360–364 at Kentron Management Science, 343 for product design and market share problems, 337–341 types of models for, 319–321 0-1 linear integer programs for, 325–340 Integer variables, 318, 319n binary expansion of, 340n modeling flexibility in, 341–344 Interarrival times, 564 Interior point solution procedures, 17-2n Inventory models, 17, 454 economic order quantity model, 454–463 economic production lot size model, 464–467 at Kellogg Company, 180 multistage inventory planning, 489 optimal lot size formula for, 501 optimal order quantity formula for, 500–501 order-quantity, reorder point model, with probabilistic demand, 480–484 periodic review model, with probabilistic demand, 484–488 planned shortages in, 467–472 in production and inventory applications, 283–286 quantity discounts in, 472–474 simulation in, 543, 558–562, 592–594 single-period, with probabilistic demand, 474–480 Inventory position, 460 Inversions of matrices, 785–786 Investments, 146–147 index funds for, 374–379 Markowitz portfolio model for, 379–382 portfolio models for, 229–235 portfolio optimization models, with transaction costs, 402–405 portfolio selection for, 161–164 simulation of financial planning application, 585–587 Iteration, 17-11, 17-15 J Jensen, Dale, 443 Jeppesen Sanderson, Inc., 173 Joint probabilities, 632 K Kellogg Company, 180 Kendall, D G., 520–521 Kentron Management Science (firm), 343 Knapsack problem, 21-10–21-16 Koopmans, Tjalling, 30 L Lead time, 461 Lead-time demand, 461 Lead-time demand distribution, 482 Lease structure applications, 164–165 Linear programming, 16, 29–30 See also Integer linear programming advertising campaign planning application, 202–203 applications of, 154 for assignment problems, 263–268 for blending problems, 183–188, 207–209 computer solutions in, 50–52 data envelopment analysis application of, 215–223 Excel for, 87–91 extreme points and optimal solution in, 48–50 financial planning applications of, 164–172 game theory and, 236–247 for goal programming problems with multiple priority levels, 674n graphical solution procedure in, 35–48 integer linear programming and, 318, 319n LINGO for, 85–87 make-or-buy decisions applications of, 164–172 marketing applications of, 154 marketing research application of, 158–160 for maximal flow problems, 279–283 for maximization, 30–34 media selection application of, 155–158 for minimization, 52–57 notation for, 62–64 portfolio models as, 229–235 portfolio selection applications of, 161–164 production scheduling applications of, 172–179, 204–205 for productivity strategy, 83–84 revenue management applications in, 223–229 858 Index Linear programming (continued) sensitivity analysis and, 94 for shortest-route problems, 276 traffic control application of, 64–65 for transportation problems, 256–263 venture capital application of, 84–85 workforce assignment applications of, 179–183, 205–207 for workload balancing, 82–83 Linear programming models, 34 for crashing, 434–436 Linear trends, 726–730 LINGO, 14n for all-integer linear programs, 324–325 extra variables in, 379n for forecasting, 759–760 free variables in, 378n for linear programming, 85–87 for nonlinear optimization problems, 408–409 for sensitivity analysis, 150–152 Little, John D C., 517 Little's flow equations, 517–518 Local optima, 371–374 Location problems, 334–337 Long-term disability claims, 612–613 Lot size, 464 Lot size model, 464–467 optimal lot size formula for, 501 LP relaxation, 320, 322 Lustig, Irv, M Machine repair problems, 528–529 Make-or-buy decisions, 164–172 Management science techniques, 2, 16–18 Markov process models, 17–18 at Merrill Lynch, 13 Marathon Oil Company, 154 Marginal costs, 15 Marketing applications of linear programming, 154 advertising campaign planning, 202–203 forecasting adoption of new products, 387–391 marketing planning, 154 marketing research, 158–160 media selection, 155–158 Marketing research, 158–160 Market share analysis of, 763–770 and product design problems, 337–341 Markov chains with stationary transition probabilities, 762 Markov decision processes, 770n Markov process models, 18, 762, 17–18 for accounts receivable analysis, 771–775 blackjack application of, 781–782 curling application of, 776 for market share analysis, 763–770 matrix inversions for, 785–786 matrix notation for, 782–783 matrix operations for, 783–785 Markowitz, Harry, 379 Markowitz portfolio model, 379–382 Mathematical models, 7, 34 Mathematical programming models, 34n Matrices Excel for inversions of, 785–786 fundamental matrices, 772–774 notation for, 782–783 operations of, 783–785 Maximal flow problems, 279–283 Maximization graphical solution procedure for, 46 linear programming for, 30–34 local and global maximums, 371 in maximal flow problems, 279–283 in transportation problems, 261 in TreePlan, 658 MeadWestvao Corporation, 29 Mean absolute error (MAE), 714 for moving averages, 718–720 Mean absolute percentage error (MAPE), 715, 716 for moving averages, 718–720 Means horizontal patterns fluctuating around, 705 moving averages, 717–720 Mean squared error (MSE), 715–716 exponential smoothing and, 724 for moving averages, 718–720 Media selection applications, 155–158 Medical applications drug decision analysis in, 629 for health care services, 762 hospital performance, 216–222 medical screening test, 634 Memoryless property, 770n Merrill Lynch, 13 Microsoft Office Project, 450–452 Middleton, Michael R., 653n Minimal spanning tree algorithm, 20-2–20-5 Minimax regret approach, 608–610 Minimization, 18-17 linear programming for, 52–57 local and global minimums, 371 in transportation problems, 261–263 in TreePlan, 658 Minimum cost method, 19-5–19-6 Minimum ratio test, 17-11 Mixed-integer linear programs, 318, 320, 340n at Kentron Management Science, 343 Mixed strategy solutions, 239–246 Model development, 7–10 Modeling (problem formulation), 31–33 flexibility in, with 0-1 linear integer programs, 341–344 Models, for assignment problems, 267 of cost, revenue, and profits, 14–16 in goal programming, 667 859 Index for integer linear programming, 319–321 portfolio models, 229–235 scoring models, 674–678 for shortest-route problems, 279 for transportation problems, 262 for transshipment problem, 274–275 Model solutions, 11–12 Modified distribution method (MODI), 19-7 Monte Carlo simulations, 557n Monthly data, for seasonality, 739–740 Moving averages, 717–720 Excel for, 753 exponential smoothing of, 721–725 moving, 720 Multicriteria decision making, 660 analytic hierarchy process for, 679–680 establishing priorities in, using AHP, 680–688 Excel for scoring models in, 701–702 goal programming in, 660–668 for multiple goals at same priority level, 668–674 problems in, scoring models for, 674–678 Multiple-channel waiting line models, 512 with Poisson arrivals, arbitrary service times, and no waiting lines, 524–526 with Poisson arrivals and exponential service times, 512–517 Multiple-choice constraints, 341–342 Multistage inventory planning, 489 Mutual funds, 229–235 Mutually exclusive constraints, 341–342 N Naïve forecasting method, 713 National Car Rental, 224, 479–480 Net evaluation index, 19-8 Net evaluation rows, 17-3 Netherlands, 484, 576–577 Network flow problems, 256 assignment problems, 263–268 at AT&T, 283 maximal flow problems, 279–283 shortest-route problems, 276–279 transportation problems, 256–263 transshipment problem, 268–275 Network models, 17 Networks, 257 New Haven (Connecticut) Fire Department, 530 Nodes, in networks, 257 in influence diagrams, 605 in project networks, 414 in shortest-route problems, 276 transshipment nodes, 268 Non-basis variables, 17-4 Nonlinear optimization problems, 366, 392 blending problems, 382–387 Excel for, 409–411 for forecasting adoption of new products, 387–391 index fund construction as, 374–379 LINGO for, 408–409 portfolio optimization models, with transaction costs, 402–405 production application of, 367–374 for scheduling lights and crews, 366–367 Nonlinear programming, 17 Nonlinear trends, 730–733 Normal distributions, 429 Normalized pairwise comparison matrix, 684 North American Product Sourcing Study, 275 Notation for linear programming, 62–64 matrix notation, 782–783 Nutrition Coordinating Center (NCC; University of Minnesota), 116 O Objective function coefficients, 18-2–18-6 Objective functions, coefficients for, 95–100, 103n in goal programming, 663 for multiple goals at same priority level, 670–671 in nonlinear optimization problems, 366 Oglethorpe Power Corporation, 635 Ohio Edison (firm), 603 100 percent rule, 18-12n Operating characteristics, 503 in multiple-channel waiting line model, with Poisson arrivals and arbitrary service times, 524–526 in multiple-channel waiting line model, with Poisson arrivals and exponential service times, 513–517 in single-channel waiting line model, with Poisson arrivals and arbitrary service times, 521–523 in single-channel waiting line model, with Poisson arrivals and exponential service times, 508–510 for waiting line models with finite calling populations, 527–529 Operations management applications blending problems, 183–188, 207–209 make-or-buy decisions, 164–172 production scheduling, 172–179, 204–205 workforce assignments, 179–183, 205–207 Operations research, 18 Opportunity losses, 608, 614–615n, 19-23 Optimality criterion, 17-18 Optimal lease structure, 164–165 Optimal lot size formula, 501 Optimal order quantity formula, 500–501 Optimal solutions, 11, 43–44, 48–50 alternative, 57–58 infeasibility and, 58–60 local and global, in nonlinear optimization problems, 371–374 sensitivity analysis and, 93 860 Index Optimistic approach, in decision making, 607 Ordering costs, 455, 456, 460, 464 Order-quantity, reorder point inventory model, with probabilistic demand, 480–484 Outgoing arcs, 19-8–19-9 Outputs, of simulations, 543 Overbooking, by airlines, 543 P Pairwise comparison matrix, 682–683 normalized, 684 Pairwise comparisons, in analytic hierarchy process, 681–683, 687–688 consistency in, 685–686 normalization of, 684 Parameters, 545 Paths, in networks, 416 Payoffs, in TreePlan, 656–657 Payoff tables, 605–606 Peak loads, 705 Perfect information, 613–615 Performance, operating characteristics of, 503 Performance Analysis Corporation, 102 Periodic review inventory model, with probabilistic demand, 484–488 Periodic review inventory systems, 485, 487–488 Petroleum industry, 704–705 simulation used by, 562–563 Pfizer (firm), 557 Pharmaceutical industry CVS Corporation in, 554 drug decision analysis in, 629 simulation used in, 557 Pharmacia (firm), 554 Pivot columns, 17-12 Pivot elements, 17-12 Pivot rows, 17-12 Poisson probability distributions, for waiting line problems, 505 in multiple-channel model, 512–517 in single-channel model, with arbitrary service times, 521–523 in single-channel model, with exponential service times, 508–512 Pooled components, 382 Pooling problems, 382–387 Porsche Shop, 443–444 Portfolio optimization models, 229–235 Markowitz portfolio model, 379–382 with transaction costs, 402–405 Portfolio selection, 161–164 in simulation of financial planning application, 585–587 Posterior probabilities, 620, 633 Postoptimality analysis, 93 See also Sensitivity analysis Preboard airport screening simulation, 573 Preemptive priorities, 661, 663 Primal problems, 18-14, 18-17 finding dual of, 18-18–18-20 using duals to solve, 18-18 Principle of optimality, 21-2 Priority level and goals, 661 multiple goals at same priority level, 668–674 problems with one priority level, 673n Prior probabilities, 620 Probabilistic demand models order-quantity, reorder point inventory model, 480–484 periodic review inventory model, 484–488 single-period inventory model with, 474–480 Probabilistic inputs, 543 generating values for, 549–554 Probabilistic models, 10 Probabilities conditional, 632 in TreePlan, 656–657 Problem formulation (modeling), 31–33, 604–605 decision trees in, 606–607 influence diagrams in, 605 payoff tables in, 605–606 Process design problem, 21-26–21-27 Proctor & Gamble, 275–276 Product design and market share problems, 337–341 forecasting adoption of new products, 387–391 Product development, simulation in, 543 Production application of nonlinear optimization problems for, 367–374 blending problems in, 183–188 at Kellogg Company, 180 production and inventory applications, 283–286 scheduling problems for, 172–179, 204–205, 359–360 Production lot size model, 464–467 optimal lot size formula for, 501 Productivity, linear programming for, 83–84 Product Sourcing Heuristic (PSH), 275–276 Profits, models of, 15 Program Evaluation and Review Technique (PERT), 17, 413 for project scheduling with known activity times, 413–422 Project networks, 413 Project scheduling, 413 with known activity times, 413–422 Microsoft Office Project for, 450–452 PERT/CPM, 17 time-cost tradeoffs in, 431–436 with uncertain activity times, 422–430 Proportionality, 34n Pseudorandom numbers, 549n 861 Index Pure network flow problems, 275n Pure strategy solutions, 238–239 Q Quadratic trend equation, 731 Qualitative forecasting methods, 704 Quantitative analysis, 6–13 in decision making, 4–6 at Merrill Lynch, 13 Quantitative forecasting methods, 704 Quantities, in inventory models optimal order quantity formula, 500–501 in order-quantity, reorder point model with probabilistic demand, 481–482 Quantity discounts, 472–474 Queue discipline, 507 Queueing theory, 503 Queues, 503 Queuing models See Waiting line models R Railroads, 263 Random numbers, 549–554 Ranges of feasibility, 106, 18-9 Ranges of optimality, 95, 18-2 Redden, Emory F., 268 Reduced cost, 105 Redundant constraints, 48 Regression analysis, 730n Relevant costs, 106 Reorder points, 460 in order-quantity, inventory model with probabilistic demand, 480–484 Repair problems call centers and, 544 machine repair problems, 528–529 office equipment repairs, 540–541 Reports, generation of, 12 Reservations, by airlines simulation of overbooking in, 543 waiting line problems in, 539–540 Return function rn(xn, dn), 21-9 Revenue, models of, 15 Revenue management airline industry's use of, 224–230 National Car Rental's use of, 224 Review periods, 485–486, 488 Risk analysis, 545 in conservative portfolios, 230–232 in decision analysis, 615–616 example of, 554–556 in moderate portfolio, 232–235 in pharmaceutical industry, 557 simulation in, 547–554, 590–592 what-if analysis, 545–547 Risk profiles, 627–629 Rounding, in all-integer linear programs, 322–323 S Saaty, Thomas L., 679 Saddle points, 239 Safety stocks, 483 Sample information, 620 expected values of, 629–630 Satellite television, 391–392 Scheduling See also Project scheduling crew scheduling problem, 318–319 of flights and crews, 340–341 production scheduling, 172–179, 204–205, 359–360 project scheduling, 413 project scheduling, PERT/CPM for, 17 volunteer scheduling problem, 340–341 Scoring models, 674–678 Excel for, 701–702 at Ford Motor Company, 678 used in analytic hierarchy process, 689n Seasonality and seasonal patterns, 709–710, 733 monthly data for, 739–740 trends and, 710–712, 737–739 without trends, 734–737 Seasongood & Mayer (firm), 422 Sensitivity analysis, 93–95 cautionary note on, 344 computer solutions to, 103–110, 118–122 in decision analysis, 615–620 for economic order quantity model, 461–462 Excel for, 148–150 graphical approach to, 95–103 limitations of, 110–115 LINGO for, 150–152 Service level, 484 Service rates, 523 Service times, 523 Setup costs, 464 Shadow price See Dual values Shortages, in inventory models, 467–472 Shortest-route problems, 276–279 Short-term disability claims, 612–613 Simon, Herbert A., Simplex-based sensitivity analysis and duality duality, 18-14–18-20 sensitivity analysis with Simples tableu, 18-2–18-13 Simplex method, 2, 17-2 algebraic overview of, 17-2–17-5, 17-32–17-33 assignment problems, 19-18–19-24 basic feasible solutions, 17-4–17-5 basic solution, 17-4 criterion for removing a variable from the current basis (minimum ratio test), 17-11 degeneracy, 17-33–17-35 equality constraints, 17-24–17-25 general case tableau form, 17-20–17-27 greater-than-or-equal-to constraints, 17-20–17-24 improving the solution, 17-10–17-12 infeasibility, 17-29–17-31, 17-35n 862 Index Simplex method (continued) interpreting optimal solutions, 17-15–17-19 interpreting results of an iteration, 17-15, 17-15–17-18 phase I of, 17-23 phase II of, 17-23–17-24 simplex tableau, 17-7–17-10 unboundedness, 17-31–17-32 Simplex tableau, 17-7–17-10 calculating next, 17-12–17-14 criterion for entering a new variable into Basis, 17-11 Simulation experiments, 543 Simulations, 17, 543–544 advantages and disadvantages of use of, 575–576 computer implementation of, 574–575 Crystal Ball for, 597–601 of drive-through waiting lines, 589–590 Excel for, 590–597 of financial planning application, 585–587 of golf course tee time reservations, 587–589 inventory simulations, 558–562 preboard airport screening simulation, 573 in risk analysis, 547–556 used by petroleum industry, 562–563 verification and validation issues in, 575 of waiting line models, 563–574 warehouse efficiency simulation, 576–577 Simultaneous changes, 111–112, 18-13 Single-channel waiting lines, 504–507 with Poisson arrivals and arbitrary service times, 521–523 with Poisson arrivals and exponential service times, 508–512 Single-criterion decision problems, Single-period inventory models with probabilistic demand, 474–480 Slack variables, 47–48 Smoothing, exponential, 721–725 Sousa, Nancy L S., 603 Spanning trees, 20-2 Spreadsheets See also Excel for exponential smoothing, 726n for simulations, 574–575 Stage transformation function, 21-8 Standard form, in linear programming, 47, 48 State of the system, 763 State probabilities, 765 States of nature, 604 sample information on, 620 State variables xn and xnϪ1, 21-8 Static simulation models, 563 Steady-state operation, 507–508 Steady-state probabilities, 768 Stepping-stone method, 19-8–19-12 Stochastic (probabilistic) models, 10 See also Probabilistic demand models Stockouts (shortages), 467 Strategies mixed, 239–246 pure, 238–239 Sunk costs, 106 Supply constant supply rate assumption for, 464 quantity discounts in, 472–474 in transportation problems, 260 Surplus variables, 55–56 Synthesization, 684–685 System constraints, 667n T Tableau form, 17-5–17-7 eliminating negative right-hand-side values, 17-25–17-26 equality constrains, 17-24–17-25 general case, 17-20–17-27 steps to create, 17-26–17-27 Target values, 661 Taylor, Frederic W., Tea production, 122–123 Telecommunications AT&T, 283 satellite television, 391–392 Time series analysis, 704 exponential smoothing in, 721–725 moving averages for, 717–720 spreadsheets for, 726n Time series patterns, 705 cyclical patterns, 713 horizontal patterns, 705–707 linear trend patterns, 726–730 nonlinear trend patterns, 730–733 seasonal patterns, 709–710, 733–740 trend patterns, 707–709 trend patterns, seasonal patterns and, 710–712 Timing See Activity times; Project scheduling Tornado diagrams, 620n Total balance method, for accounts receivable, 771 Total cost models, 455n, 465–467 Traffic control, 64–65 simulation in, 544 Transaction costs, 402–405 Transient periods, 507–508 Transition probabilities, 763–764 Transportation problems, 256–263, 19-2 assignment problem as special case of, 267n Excel for, 308–310 shortest-route problems, 276–279 transshipment problem and, 268–275 for Union Pacific Railroad, 263 Transportation simplex method, 19-17 phase I: finding initial feasible solution, 19-2–19-6 phase II: iterating to optimal solution, 19-7–19-16 problem variations, 19-17–19-18 863 Index Transportation tableaux, 19-2 Transshipment problems, 268–275 Excel for, 313–315 production and inventory applications of, 283–286 TreePlan (software), 653–658 Trend patterns, 707–709 Excel for projection of, 754 linear, 726–730 nonlinear, 730–733 seasonality and, 734–737 seasonal patterns and, 710–712 Trials of the process, 763 Truck leasing, 147–148 Two-person, zero-sum games, 236, 247n U Unbounded feasible region, 17-31–17-32, 17-35n Unbounded solutions, 60–61, 61n Unconstrained nonlinear optimization problems, 367–378 Union Pacific Railroad, 263 Unit columns, 17-8 Unit vectors, 17-8 Upjohn (firm), 554 Utility industry, 704–705 V Validation issues, in simulations, 575 Valley Metal Container (VMC), 320–321 Values expected values, 610 random, generating, 549–554 in simulation experiments, 543 Vancouver International Airport, 573 Vanguard Index Funds, 375 Variable annuities, 236 Variable costs, 14 Variables binary variables, 318 decision variables, definitional variables, 234n free variables, 378n integer variables, 318 slack variables, 47–48 surplus variables, 55–56 Venture capital, 84–85 Verification issues, in simulations, 575 Volume, models of, 14–15 Volunteer scheduling problem, 340–341 W Waiting line (queueing) models, 17, 503 for airline reservations, 539–540 for automatic teller machines, 503–504 drive-through line simulation of, 589–590 economic analysis of, 519–520 Excel for, 511–512 with finite calling populations, 526–529 Little's flow equations for, 517–518 multiple-channel, with Poisson arrivals, arbitrary service times and no waiting line, 524–526 multiple-channel, with Poisson arrivals and exponential service times, 512–517 office equipment repairs, 540–541 other models, 520–521 preboard airport screening simulation, 573 simulation in, 544, 563–574 single-channel, with Poisson arrivals and arbitrary service times, 521–523 single-channel, with Poisson arrivals and exponential service times, 508–512 structures of, 504–508 Warehouse efficiency, 576–577 Water quality management problem, 21-21 Weighted moving averages, 720 exponential smoothing of, 721–725 Welte Mutual Funds, Inc., 161–164 What-if analysis, 545–547 Whole Foods Markets, 507 Workers' Compensation Board of British Columbia, 612–613 Workforce assignment applications, 179–183, 205–207 Workload balancing, 82–83 Worst-case scenarios, 546 Z 0-1 linear integer programs, 318, 320, 325–340 for bank location problems, 334–337 binary expansion of integer variables for, 340n for capital budgeting problems, 325–329 for crew scheduling problem, 318–319 for distribution system design problems, 329–334 at Kentron Management Science, 343 multiple-choice and mutually-exclusive constraints in, 341–342 for product design and market share problems, 337–341 for volunteer scheduling problem, 340–341 Zero-sum games, 236, 247n David R Anderson Dennis J Sweeney Thomas A Williams Jeffrey D Camm Kipp Martin Dear Colleague: to Management Science the 13th Edition of An Introduction of sion revi the ent pres to sed plea We are ld like to share some of the meet your teaching needs, and we wou will on editi new the that in certa We are changes in the new edition are adding a new member to ges, we want to announce that we Prior to getting into the content chan University He has been at the received his Ph.D from Clemson the author team: Jeffrey Camm Jeff ford University and a visiting Stan a visiting scholar at been has and , 1984 e sinc ti inna Cinc University of th College Jeff has published Tuck School of Business at Dartmou the at n ratio inist adm ness busi of r professo in operations management At of optimization applied to problems area ral gene the in rs pape 30 than more Excellence and in 2006 received named the Dornoff Fellow of Teaching was he ti, inna Cinc of y ersit Univ the currently serves as editor-in-chief of Operations Research Practice He the INFORMS Prize for the Teaching Education We welcome Jeff to the board of INFORMS Transactions on of Interfaces, and is on the editorial years to come s will make the text even better in the author team and expect his new idea kplace Because it is a ® inant analysis software used in the wor dom the me beco has el Exc that r It is clea and engineering graduates must be l decision models, today’s business ytica anal ding buil for tool l erfu pow basics and modeling to this edition e added an appendix on spreadsheet efor ther have We el Exc in t cien profi el, and how to audit the model how to build a reliable spreadsheet mod Appendix A covers basic Excel skills, once it is constructed lysis and Interpretation (Linear Programming: Sensitivity Ana pter Cha sed revi ly cant ifi sign We have focus of this chapter, but we ysis and its interpretation are still the anal ty itivi sens al ition Trad ) tion of Solu example, difficulty with multiple of traditional sensitivity analysis (for ns tatio limi the on rial mate d adde have to explore models by actually ficients) and we encourage students changes and changes in constraint coef ents that a model should be viewed problems Indeed, we teach our stud changing the data and re-solving the the data for multiple scenarios of experimentation; this means running as a laboratory It should be used for the input data e major revisions and updates and Forecasting) has also undergon lysis Ana es Seri e (Tim 15 pter Cha to select an appropriate forecasting on using the pattern in a time series s focu d ease incr has now 15 pter Cha moving averages and exponential sure forecast accuracy, we show how method After discussing how to mea We show how to use optimization time series with a horizontal pattern smoothing can be used to forecast a ng Then, for time series that othing constant in exponential smoothi to find the best-fitting value of the smo to set up and solve the least a curve-fitting approach, showing how have only a long-term trend, we take inear trend We continue the nonl parameter values for both linear and squares problem to find the best-fitting s can be used to forecast a able vari show how the use of dummy and , data onal seas for oach appr curve-fitting forces the material presented ve taking a curve-fitting approach rein belie We cts effe onal seas with s time serie in optimization chapters many of you have used The ge in terms of software We know that Finally, we have made a major chan y years With this edition, we have has accompanied the text for so man Management Scientist software that continued decided to discontinue use of The Man agement Scientist We suggest that past users of The Management Scientist move to either Excel Solv er or LINGO as a replacement App endices describing the use of Excel Solver and LINGO are provided This edition marks our transition from Excel 2007 to Excel 2010 In particular, you will find the screen shot s of Excel Solver in the appendices base d on the Solver that ships with 2010 Fortunately, there is no chan ge in how you build models using the 2010 version of Excel Solver developed by Frontline Systems How ever, those familiar with the 2007 vers ion of Exc el Solver will notice changes in the Dialog Boxes The scre en shots and corresponding discussi on we provide in the appendices will equip students to use the 2010 vers ion Considerable deliberation went into the decision to discontinue the use of The Management Scientist and there are three reasons why we are mak ing this move First, The Management Scientist software is no longer being developed and supported by its authors We not think it is beneficial to our readers to expend effort learning and using software that is no longer supported Second, students are far more likely to encounter Excel-based software in the workpla ce Finally, for optimization problems , The Management Scientist often required a fair amount of algebraic manipulation of the model to get all variables on the left-hand side of the constraints and a single constant on the right-hand side Modern softw are packages, such as LINGO and Excel Solver, not require this and allow the user to enter a model in its more natu ral formulation Users who liked the simple input format of The Management Scientist and not wan t to switch to Excel Solver may wish to use LINGO This allows for directly entering the objective function and constraints It is possible to use LINGO as your calculator and avoi d any arithmetic or algebraic simplifi cations We believe this strengthens our model-focused approach, as it eliminates the distraction of having to manipulate the model before solving At the publication Website we provide a documented LINGO model for every optimization example developed in the text We also provide an Excel Solver model for all of thes e problems For project management, a trial version of Microsoft Proj ect Professional 2010 is packaged with each new copy of the text and we include an appendix on how to use it in Chapter For these reasons, The Management Scientist software is no longer discussed in the text For those of you wishing to continue to use The Management Scientist, it is available at no extra charge on the Website for this book To access addi tional course materials, please visit www.cengagebrain.com At the Cen gageBrain.com home page, fill in the ISBN of your title (from the back cover of your book) using the search box at the top of the page This will take you to the product page where these resources can be found The focus of the text has always been on modeling and the applications of thes e models As such, the book has always been software-independent With the decision to discontinue The Management Science software, we were faced with the decision of how to present optimization output Rather than choose LINGO or Excel Solver output (which present sensitivi ty analysis in slightly different ways), we decided to present a generic output for optimization problems in the body of the chapters This removes any dependence on a single piece of software In this edition, we use the dual value rather than the dual price The dual value is defined as the change in the optimal objective func tion value resulting from an increase of one unit in the right-hand side of a constraint Using the dual valu e eliminates the need to discuss diffe rences in interpretation between maximization problem and a minimiz ation problem The dual value and its sign are interpreted the same, regardless of whether the problem is a minimization or a maximization In this generic output, we of course also present the allowable changes to the right-hand sides for which the dual value holds The new edition continues our long tradition of writing a text that is appl icati ons oriented and pragmatic We thank you for your interest in our text Our ultimate goal is to prov ide you with material that truly helps your students learn and also mak es your job as their instructor easier We wish you and your students the very best Sincerely, David R Anderson Dennis J Sweeney Thomas A Williams Jeffrey D Camm Kipp Martin