《运筹学 Operations Research》课程教学资源（书籍教材）《运筹学：应用和算法》电子书（第四版）Operations Research, Applications and Algorithms - Wayne L. Winston，4th

of the operations research study,the analyst may find that the organization does not ap- prove of the recommendation.This may result from incorrect definition of the organiza- tion's problems or from failure to involve the decision maker from the start of the project. In this case,the operations researcher should return to step 1,2,or 3. Step 7:Implement and Evaluate Recommendations If the organization has accepted the study,then the analyst aids in implementing the recommendations.The system must be constantly monitored (and updated dynamically as the environment changes)to ensure that the recommendations enable the organization to meet its objectives. In what follows,we discuss three successful management science applications.We will give a detailed(but nonquantitative)description of each application.We will tie our discus- sion of each application to the seven-step model-building process described in Section 1.2. 1.3 CITGO Petroleum Klingman et al.(1987)applied a variety of management-science techniques to CITGO Pe- troleum.Their work saved the company an estimated $70 million per year.CITGO is an oil-refining and-marketing company that was purchased by Southland Corporation(the owners of the 7-Eleven stores).We will focus on two aspects of the CITGO team's work: 1 a mathematical model to optimize operation of CITGO's refineries,and 2 a mathematical model-supply distribution marketing(SDM)system-that was used to develop an 11-week supply,distribution,and marketing plan for the entire business. Optimizing Refinery Operations Step 1 Klingman et al.wanted to minimize the cost of operating CITGO's refineries. Step 2 The Lake Charles,Louisiana,refinery was closely observed in an attempt to es- timate key relationships such as: 1 How the cost of producing each of CITGO's products (motor fuel,no.2 fuel oil,tur- bine fuel,naptha,and several blended motor fuels)depends on the inputs used to produce each product. 2 The amount of energy needed to produce each product.This required the installation of a new metering system 3 The yield associated with each input-output combination.For example,if 1 gallon of crude oil would yield.52 gallons of motor fuel,then the yield would equal 52%. 4 To reduce maintenance costs,data were collected on parts inventories and equipment breakdowns.Obtaining accurate data required the installation of a new database-management system and integrated maintenance-information system.A process control system was also installed to accurately monitor the inputs and resources used to manufacture each product. Step 3 Using linear programming(LP),a model was developed to optimize refinery op- erations.The model determines the cost-minimizing method for mixing or blending to- gether inputs to produce desired outputs.The model contains constraints that ensure that inputs are blended so that each output is of the desired quality.Blending constraints are discussed in Section 3.8.The model ensures that plant capacities are not exceeded and al-

of the operations research study, the analyst may find that the organization does not ap￾prove of the recommendation. This may result from incorrect definition of the organiza￾tion’s problems or from failure to involve the decision maker from the start of the project. In this case, the operations researcher should return to step 1, 2, or 3. Step 7: Implement and Evaluate Recommendations If the organization has accepted the study, then the analyst aids in implementing the recommendations. The system must be constantly monitored (and updated dynamically as the environment changes) to ensure that the recommendations enable the organization to meet its objectives. In what follows, we discuss three successful management science applications. We will give a detailed (but nonquantitative) description of each application. We will tie our discus￾sion of each application to the seven-step model-building process described in Section 1.2. 1.3 CITGO Petroleum Klingman et al. (1987) applied a variety of management-science techniques to CITGO Pe￾troleum. Their work saved the company an estimated $70 million per year. CITGO is an oil-refining and -marketing company that was purchased by Southland Corporation (the owners of the 7-Eleven stores). We will focus on two aspects of the CITGO team’s work: 1 a mathematical model to optimize operation of CITGO’s refineries, and 2 a mathematical model—supply distribution marketing (SDM) system—that was used to develop an 11-week supply, distribution, and marketing plan for the entire business. Optimizing Refinery Operations Step 1 Klingman et al. wanted to minimize the cost of operating CITGO’s refineries. Step 2 The Lake Charles, Louisiana, refinery was closely observed in an attempt to es￾timate key relationships such as: 1 How the cost of producing each of CITGO’s products (motor fuel, no. 2 fuel oil, tur￾bine fuel, naptha, and several blended motor fuels) depends on the inputs used to produce each product. 2 The amount of energy needed to produce each product. This required the installation of a new metering system. 3 The yield associated with each input–output combination. For example, if 1 gallon of crude oil would yield .52 gallons of motor fuel, then the yield would equal 52%. 4 To reduce maintenance costs, data were collected on parts inventories and equipment breakdowns. Obtaining accurate data required the installation of a new database-management system and integrated maintenance-information system. A process control system was also installed to accurately monitor the inputs and resources used to manufacture each product. Step 3 Using linear programming (LP), a model was developed to optimize refinery op￾erations. The model determines the cost-minimizing method for mixing or blending to￾gether inputs to produce desired outputs. The model contains constraints that ensure that inputs are blended so that each output is of the desired quality. Blending constraints are discussed in Section 3.8. The model ensures that plant capacities are not exceeded and al- 6 CHAPTER 1 An Introduction to Model Building

lows for the fact that each refinery may carry an inventory of each end product.Sections 3.10 and 4.12 discuss inventory constraints. Step 4 To validate the model,inputs and outputs from the Lake Charles refinery were collected for one month.Given the actual inputs used at the refinery during that month, the actual outputs were compared to those predicted by the model.After extensive changes,the model's predicted outputs were close to the actual outputs. Step 5 Running the LP yielded a daily strategy for running the refinery.For instance,the model might,say,produce 400,000 gallons of turbine fuel using 300,000 gallons of crude 1 and 200,000 gallons of crude 2. Steps 6 and 7 Once the database and process control were in place,the model was used to guide day-to-day refinery operations.CITGO estimated that the overall benefits of the refinery system exceeded $50 million annually. The Supply Distribution Marketing (SDM)System Step 1 CITGO wanted a mathematical model that could be used to make supply,distri- bution,and marketing decisions such as: 1 Where should crude oil be purchased? 2 Where should products be sold? 3 What price should be charged for products? 4 How much of each product should be held in inventory? The goal,of course,was to maximize the profitability associated with these decisions. Step 2 A database that kept track of sales,inventory,trades,and exchanges of all refined products was installed.Also,regression analysis (see Chapter 24)was used to develop forecasts for wholesale prices and wholesale demand for each CITGO product. Steps 3 and 5 A minimum-cost network flow model(MCNFM)(see Section 7.4)is used to determine an 11-week supply,marketing,and distribution strategy.The model makes all decisions mentioned in step 1.A typical model run that involved 3,000 equations and 15,000 decision variables required only 30 seconds on an IBM 4381. Step 4 The forecasting modules are continuously evaluated to ensure that they continue to give accurate forecasts. Steps 6 and 7 Implementing the SDM required several organizational changes.A new vice-president was appointed to coordinate the operation of the SDM and LP refinery model.The product supply and product scheduling departments were combined to im- prove communication and information flow. 1.4 San Francisco Police Department Scheduling Taylor and Huxley (1989)developed a police patrol scheduling system(PPSS).All San Francisco(SF)police precincts use PPSS to schedule their officers.It is estimated that PPSS saves the SF police more than S5 million annually.Other cities such as Virginia

lows for the fact that each refinery may carry an inventory of each end product. Sections 3.10 and 4.12 discuss inventory constraints. Step 4 To validate the model, inputs and outputs from the Lake Charles refinery were collected for one month. Given the actual inputs used at the refinery during that month, the actual outputs were compared to those predicted by the model. After extensive changes, the model’s predicted outputs were close to the actual outputs. Step 5 Running the LP yielded a daily strategy for running the refinery. For instance, the model might, say, produce 400,000 gallons of turbine fuel using 300,000 gallons of crude 1 and 200,000 gallons of crude 2. Steps 6 and 7 Once the database and process control were in place, the model was used to guide day-to-day refinery operations. CITGO estimated that the overall benefits of the refinery system exceeded $50 million annually. The Supply Distribution Marketing (SDM) System Step 1 CITGO wanted a mathematical model that could be used to make supply, distri￾bution, and marketing decisions such as: 1 Where should crude oil be purchased? 2 Where should products be sold? 3 What price should be charged for products? 4 How much of each product should be held in inventory? The goal, of course, was to maximize the profitability associated with these decisions. Step 2 A database that kept track of sales, inventory, trades, and exchanges of all refined products was installed. Also, regression analysis (see Chapter 24) was used to develop forecasts for wholesale prices and wholesale demand for each CITGO product. Steps 3 and 5 A minimum-cost network flow model (MCNFM) (see Section 7.4) is used to determine an 11-week supply, marketing, and distribution strategy. The model makes all decisions mentioned in step 1. A typical model run that involved 3,000 equations and 15,000 decision variables required only 30 seconds on an IBM 4381. Step 4 The forecasting modules are continuously evaluated to ensure that they continue to give accurate forecasts. Steps 6 and 7 Implementing the SDM required several organizational changes. A new vice-president was appointed to coordinate the operation of the SDM and LP refinery model. The product supply and product scheduling departments were combined to im￾prove communication and information flow. 1.4 San Francisco Police Department Scheduling Taylor and Huxley (1989) developed a police patrol scheduling system (PPSS). All San Francisco (SF) police precincts use PPSS to schedule their officers. It is estimated that PPSS saves the SF police more than $5 million annually. Other cities such as Virginia 1.4 San Francisco Police Department Scheduling 7

Beach,Virginia,and Richmond,California,have also adopted PPSS.Following our seven- step model-building procedure,here is a description of PPSS. Step 1 The SFPD wanted a method to schedule patrol officers in each precinct that would quickly produce (in less than one hour)a schedule and graphically display it.The program should first determine the personnel requirements for each hour of the week.For example,38 officers might be needed between I A.M.and 2 A.M.Sunday but only 14 of- ficers might be needed from 4 A.M.to 5 A.M.Sunday.Officers should then be scheduled to minimize the sum over each hour of the week of the shortages and surpluses relative to the needed number of officers.For example,if 20 officers were assigned to the mid- night to 8 A.M.Sunday shift,we would have a shortage of 38-20=18 officers from 1 to 2 A.M.and a surplus of 20 -14 =6 officers from 4 to 5 A.M.A secondary criterion was to minimize the maximum shortage because a shortage of 10 officers during a sin- gle hour is far more serious than a shortage of one officer during 10 different hours.The SFPD also wanted a scheduling system that precinct captains could easily fine-tune to produce the optimal schedule. Step 2 The SFPD had a sophisticated computer-aided dispatch(CAD)system to keep track of all calls for police help,police travel time,police response time,and so on.SFPD had a standard percentage of time that administrators felt each officer should be busy.Us- ing CAD,it is easy to determine the number of workers needed each hour.Suppose,for example,an officer should be busy 80%of the time and CAD indicates that 30.4 hours of work come in from 4 to 5 A.M.Sunday.Then we need 38 officers from 4 to 5 A.M.on Sunday [.8*(38)=30.4 hours]. Step 3 An LP model was formulated (see Section 3.5 for a discussion of scheduling models).As discussed in step 1,the primary objective was to minimize the sum of hourly shortages and surpluses.At first,schedulers assumed that officers worked five consecu- tive days for eight hours a day(this was the policy prior to PPSS)and that there were three shift starting times (say,6 A.M.,2 P.M.,and 10 A.M.).The constraints in the PPSS model reflected the limited number of officers available and the relationship of the num- ber of officers working each hour to the shortages and surpluses for that hour.Then PPSS would produce a schedule that would tell the precinct captain how many officers should start work at each possible shift time.For example,PPSS might say that 20 officers should start work at 6 A.M.Monday (working 6 A.M.-2 P.M.Monday-Friday)and 30 officers should start work at 2 P.M.Saturday (working 2 P.M.-10 P.M.Saturday-Wednesday).The fact that the number of officers assigned to a start time must be an integer made it far more difficult to find an optimal schedule.(Problems in which decision variables must be integers are discussed in Chapter 9.) Step 4 Before implementing PPSS,the SFPD tested the PPSS schedules against manu- ally created schedules.PPSS produced an approximately 50%reduction in both surpluses and shortages.This convinced the department to implement PPSS Step 5 Given the starting times for shifts and the type of work schedule [four consecu- tive days for 10 hours per day (the 4/10 schedule)or five consecutive days for eight hours per day(the 5/8 schedule)],PPSS can produce a schedule that minimizes the sum of short- ages and surpluses.More important,PPSS can be used to experiment with shift times and work rules.Using PPSS,it was found that if only three shift times are allowed,then a 5/8 schedule was superior to a 4/10 schedule.If,however,five shift times were allowed,then a 4/10 schedule was found to be superior.This finding was of critical importance because police officers had wanted to switch to a 4/10 schedule for years.The city had resisted 4/10 schedules because they appeared to reduce productivity.PPSS showed that 4/10 schedules need not reduce productivity.After the introduction of PPSS,the SFPD went

Beach, Virginia, and Richmond, California, have also adopted PPSS. Following our seven￾step model-building procedure, here is a description of PPSS. Step 1 The SFPD wanted a method to schedule patrol officers in each precinct that would quickly produce (in less than one hour) a schedule and graphically display it. The program should first determine the personnel requirements for each hour of the week. For example, 38 officers might be needed between 1 A.M. and 2 A.M. Sunday but only 14 of￾ficers might be needed from 4 A.M. to 5 A.M. Sunday. Officers should then be scheduled to minimize the sum over each hour of the week of the shortages and surpluses relative to the needed number of officers. For example, if 20 officers were assigned to the mid￾night to 8 A.M. Sunday shift, we would have a shortage of 38  20 18 officers from 1 to 2 A.M. and a surplus of 20  14 6 officers from 4 to 5 A.M. A secondary criterion was to minimize the maximum shortage because a shortage of 10 officers during a sin￾gle hour is far more serious than a shortage of one officer during 10 different hours. The SFPD also wanted a scheduling system that precinct captains could easily fine-tune to produce the optimal schedule. Step 2 The SFPD had a sophisticated computer-aided dispatch (CAD) system to keep track of all calls for police help, police travel time, police response time, and so on. SFPD had a standard percentage of time that administrators felt each officer should be busy. Us￾ing CAD, it is easy to determine the number of workers needed each hour. Suppose, for example, an officer should be busy 80% of the time and CAD indicates that 30.4 hours of work come in from 4 to 5 A.M. Sunday. Then we need 38 officers from 4 to 5 A.M. on Sunday [.8*(38) 30.4 hours]. Step 3 An LP model was formulated (see Section 3.5 for a discussion of scheduling models). As discussed in step 1, the primary objective was to minimize the sum of hourly shortages and surpluses. At first, schedulers assumed that officers worked five consecu￾tive days for eight hours a day (this was the policy prior to PPSS) and that there were three shift starting times (say, 6 A.M., 2 P.M., and 10 A.M.). The constraints in the PPSS model reflected the limited number of officers available and the relationship of the num￾ber of officers working each hour to the shortages and surpluses for that hour. Then PPSS would produce a schedule that would tell the precinct captain how many officers should start work at each possible shift time. For example, PPSS might say that 20 officers should start work at 6 A.M. Monday (working 6 A.M.–2 P.M. Monday–Friday) and 30 officers should start work at 2 P.M. Saturday (working 2 P.M.–10 P.M. Saturday–Wednesday). The fact that the number of officers assigned to a start time must be an integer made it far more difficult to find an optimal schedule. (Problems in which decision variables must be integers are discussed in Chapter 9.) Step 4 Before implementing PPSS, the SFPD tested the PPSS schedules against manu￾ally created schedules. PPSS produced an approximately 50% reduction in both surpluses and shortages. This convinced the department to implement PPSS. Step 5 Given the starting times for shifts and the type of work schedule [four consecu￾tive days for 10 hours per day (the 4/10 schedule) or five consecutive days for eight hours per day (the 5/8 schedule)], PPSS can produce a schedule that minimizes the sum of short￾ages and surpluses. More important, PPSS can be used to experiment with shift times and work rules. Using PPSS, it was found that if only three shift times are allowed, then a 5/8 schedule was superior to a 4/10 schedule. If, however, five shift times were allowed, then a 4/10 schedule was found to be superior. This finding was of critical importance because police officers had wanted to switch to a 4/10 schedule for years. The city had resisted 4/10 schedules because they appeared to reduce productivity. PPSS showed that 4/10 schedules need not reduce productivity. After the introduction of PPSS, the SFPD went 8 CHAPTER 1 An Introduction to Model Building

to 4/10 schedules and improved productivity!PPSS also enables the department to exper- iment with a mix of one-officer and two-officer patrol cars. Steps 6 and 7 It is estimated that PPSS created an extra 170,000 productive hours per year,thereby saving the city of San Francisco S5.2 million per year.Ninety-six percent of all workers preferred PPSS generated schedules to manually generated schedules.PPSS enabled SFPD to make strategic changes(such as adopting the 4/10 schedule),which made officers happier and increased productivity.Response times to calls improved by 20%after PPSS was adopted. A major reason for the success of PPSS was that the system allowed precinct captains to fine-tune the computer-generated schedule and obtain a new schedule in less than one minute.For example,precinct captains could easily add or delete officers and add or delete shifts and quickly see how these changes modified the master schedule. 1.5 GE Capital GE Capital provides credit card service to 50 million accounts.The average total out- standing balance exceeds $12 billion.GE Capital,led by Makuch et al.(1989),developed the PAYMENT system to reduce delinquent accounts and the cost of collecting from delinquent accounts. Step 1 At any one time,GE Capital has more than $1 billion in delinquent accounts. The company spends $100 million per year processing these accounts.Each day,workers contact more than 200,000 delinquent credit card holders with letters,messages,or live calls.The company's goal was to reduce delinquent accounts and the cost of processing them.To do this,GE Capital needed to come up with a method of assigning scarce labor resources to delinquent accounts.For example,PAYMENT determines which delinquent accounts receive live phone calls and which delinquent accounts receive no contact. Step 2 The key to modeling delinquent accounts is the concept of a delinquency move- ment matrix(DMM).The DMM determines how the probability of the payment on a delinquent account during the current month depends on the following factors:size of un- paid balance (either <$300 or =S300),action taken (no action,live phone call,taped message,letters),and a performance score (high,medium,or low).The higher the per- formance score associated with a delinquent account,the more likely the account is to be collected.Table 1 lists the probabilities for a $250 account that is two months delinquent, has a high performance score,and is contacted with a phone message. TABLE 1 Sample Entries in DMM Event Probability Account completely paid .30 One month is paid .40 Nothing is paid 30 Because GE Capital has millions of delinquent accounts,there is ample data to accu- rately estimate the DMM.For example,suppose there were 10,000 two-month delinquent accounts with balances under $300 that have a high performance score and are contacted with phone messages.If 3,000 of those accounts were completely paid off during the cur- rent month,then we would estimate the probability of an account being completely paid off during the current month as 3,000/10,000 =.30

to 4/10 schedules and improved productivity! PPSS also enables the department to exper￾iment with a mix of one-officer and two-officer patrol cars. Steps 6 and 7 It is estimated that PPSS created an extra 170,000 productive hours per year, thereby saving the city of San Francisco $5.2 million per year. Ninety-six percent of all workers preferred PPSS generated schedules to manually generated schedules. PPSS enabled SFPD to make strategic changes (such as adopting the 4/10 schedule), which made officers happier and increased productivity. Response times to calls improved by 20% after PPSS was adopted. A major reason for the success of PPSS was that the system allowed precinct captains to fine-tune the computer-generated schedule and obtain a new schedule in less than one minute. For example, precinct captains could easily add or delete officers and add or delete shifts and quickly see how these changes modified the master schedule. 1.5 GE Capital GE Capital provides credit card service to 50 million accounts. The average total out￾standing balance exceeds $12 billion. GE Capital, led by Makuch et al. (1989), developed the PAYMENT system to reduce delinquent accounts and the cost of collecting from delinquent accounts. Step 1 At any one time, GE Capital has more than $1 billion in delinquent accounts. The company spends $100 million per year processing these accounts. Each day, workers contact more than 200,000 delinquent credit card holders with letters, messages, or live calls. The company’s goal was to reduce delinquent accounts and the cost of processing them. To do this, GE Capital needed to come up with a method of assigning scarce labor resources to delinquent accounts. For example, PAYMENT determines which delinquent accounts receive live phone calls and which delinquent accounts receive no contact. Step 2 The key to modeling delinquent accounts is the concept of a delinquency move￾ment matrix (DMM). The DMM determines how the probability of the payment on a delinquent account during the current month depends on the following factors: size of un￾paid balance (either $300 or $300), action taken (no action, live phone call, taped message, letters), and a performance score (high, medium, or low). The higher the per￾formance score associated with a delinquent account, the more likely the account is to be collected. Table 1 lists the probabilities for a $250 account that is two months delinquent, has a high performance score, and is contacted with a phone message. TAB LE 1 Sample Entries in DMM Event Probability Account completely paid .30 One month is paid .40 Nothing is paid .30 Because GE Capital has millions of delinquent accounts, there is ample data to accu￾rately estimate the DMM. For example, suppose there were 10,000 two-month delinquent accounts with balances under $300 that have a high performance score and are contacted with phone messages. If 3,000 of those accounts were completely paid off during the cur￾rent month, then we would estimate the probability of an account being completely paid off during the current month as 3,000/10,000 .30. 1.5 GE Capital 9

Step 3 GE Capital developed a linear optimization model.The objective function for the PAYMENT model was to maximize the expected delinquent accounts collected during the next six months.The decision variables represented the fraction of each type of delinquent account (accounts are classified by payment balance,performance score,and months delinquent)that experienced each type of contact(no action,live phone call,taped mes- sage,or letter).The constraints in the PAYMENT model ensure that available resources are not overused.Constraints also relate the number of each type of delinquent account present in,say,January to the number of delinquent accounts of each type present during the next month (February).This dynamic aspect of the PAYMENT model is crucial to its success.Without this aspect,the model would simply"skim"the accounts that are easi- est to collect each month.This would result in few collections during later months Step 4 PAYMENT was piloted on a $62 million portfolio for a single department store. GE Capital managers came up with their own strategies for allocating resources(collec- tively called CHAMPION).The store's delinquent accounts were randomly assigned to the CHAMPION and PAYMENT strategies.PAYMENT used more live phone calls and more "no action"than the CHAMPION strategies.PAYMENT also collected $180,000 per month more than any of the CHAMPION strategies,a 5%to 7%improvement.Note that using more of the no-action strategy certainly leads to a long-run increase in cus- tomer goodwill! Step 5 As described in step 3,for each type of account,PAYMENT tells the credit man- agers the fraction that should receive each type of contact.For example,for three-month delinquent accounts with a small(<$300)unpaid balance and high performance score, PAYMENT might prescribe 30%no action,20%letters,30%phone messages,and 20% live phone calls. Steps 6 and 7 PAYMENT was next applied to the 18 million accounts of the $4.6 billion Montgomery-Ward department store portfolio.Comparing the collection results to the same time period a year earlier,it was found that PAYMENT increased collections by $1.6 million per month(more than $19 million per year).This is actually a conservative esti- mate of the benefit obtained from PAYMENT,because PAYMENT was first applied to the Montgomery-Ward portfolio during the depths of a recession-and a recession makes it much more difficult to collect delinquent accounts. Overall,GE Capital estimates that PAYMENT increased collections by $37 million per year and used fewer resources than previous strategies. REFERENCES Klingman,D.,N.Phillips,D.Steiger,and W.Young."The Economy:A Multi-Billion Dollar Management Science Successful Deployment of Management Science Application,"Interfaces 22(1992,no.1):90-109. Throughout Citgo Corporation,"Interfaces 17 (1987, Taylor,P,and S.Huxley,"A Break from Tradition for the no.1):4-25. San Francisco Police:Patrol Officer Scheduling Using Makuch,W.,J.Dodge,J.Ecker,D.Granfors,and G.Hahn, an Optimization-Based Decision Support Tool,"Inter- "Managing Consumer Credit Delinquency in the US faces19(1989,no.1):4-24

Step 3 GE Capital developed a linear optimization model. The objective function for the PAYMENT model was to maximize the expected delinquent accounts collected during the next six months. The decision variables represented the fraction of each type of delinquent account (accounts are classified by payment balance, performance score, and months delinquent) that experienced each type of contact (no action, live phone call, taped mes￾sage, or letter). The constraints in the PAYMENT model ensure that available resources are not overused. Constraints also relate the number of each type of delinquent account present in, say, January to the number of delinquent accounts of each type present during the next month (February). This dynamic aspect of the PAYMENT model is crucial to its success. Without this aspect, the model would simply “skim” the accounts that are easi￾est to collect each month. This would result in few collections during later months. Step 4 PAYMENT was piloted on a $62 million portfolio for a single department store. GE Capital managers came up with their own strategies for allocating resources (collec￾tively called CHAMPION). The store’s delinquent accounts were randomly assigned to the CHAMPION and PAYMENT strategies. PAYMENT used more live phone calls and more “no action” than the CHAMPION strategies. PAYMENT also collected $180,000 per month more than any of the CHAMPION strategies, a 5% to 7% improvement. Note that using more of the no-action strategy certainly leads to a long-run increase in cus￾tomer goodwill! Step 5 As described in step 3, for each type of account, PAYMENT tells the credit man￾agers the fraction that should receive each type of contact. For example, for three-month delinquent accounts with a small ($300) unpaid balance and high performance score, PAYMENT might prescribe 30% no action, 20% letters, 30% phone messages, and 20% live phone calls. Steps 6 and 7 PAYMENT was next applied to the 18 million accounts of the $4.6 billion Montgomery-Ward department store portfolio. Comparing the collection results to the same time period a year earlier, it was found that PAYMENT increased collections by $1.6 million per month (more than $19 million per year). This is actually a conservative esti￾mate of the benefit obtained from PAYMENT, because PAYMENT was first applied to the Montgomery-Ward portfolio during the depths of a recession—and a recession makes it much more difficult to collect delinquent accounts. Overall, GE Capital estimates that PAYMENT increased collections by $37 million per year and used fewer resources than previous strategies. REFERENCES 10 CHAPTER 1 An Introduction to Model Building Klingman, D., N. Phillips, D. Steiger, and W. Young, “The Successful Deployment of Management Science Throughout Citgo Corporation,” Interfaces 17 (1987, no. 1):4–25. Makuch, W., J. Dodge, J. Ecker, D. Granfors, and G. Hahn, “Managing Consumer Credit Delinquency in the US Economy: A Multi-Billion Dollar Management Science Application,” Interfaces 22 (1992, no. 1):90–109. Taylor, P., and S. Huxley, “A Break from Tradition for the San Francisco Police: Patrol Officer Scheduling Using an Optimization-Based Decision Support Tool,” Inter￾faces 19 (1989, no. 1):4–24