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## COB 291

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**COB 291**Introduction to Management Science Michael E. Busing CIS/OM Program, James Madison University**What is Management Science?**• field of study that uses computers, statistics and mathematics to solve business problems. • sometimes referred to as operations research or decision science. • formerly, the field was available to only those with advanced knowledge of mathematics and computer programming languages. PC’s and spreadsheets have made the tools available to a much larger audience.**Air New Zealand, “Optimized Crew Scheduling at Air New**Zealand” The airline crew scheduling problem consists of the pairings problem involving the generation of minimum-cost pairings (sequences of duty periods) to cover all scheduled flights, and the rostering problem involving the assignment of pairings to individual crewmembers. Over the past fifteen years, eight application-specific optimization-based computer systems have been developed in collaboration with the University of Auckland to solve all aspects of the pairings and rostering processes for Air New Zealand's National and International operations. These systems have produced large savings, while also providing crew rosters that better respect crew preferences.**Federal Aviation Administration, “Ground Delay Program**Enhancements (GDPE) under Collaborative Decision Making (CDM)” When airport arrival capacity is reduced, the demand placed by arriving aircraft may be too great. In these cases a ground delay program (GDP) is used to delay flights before departure at their origin airport, keeping traffic at an acceptable level for the impacted arrival airport. However, GDPs sometimes lacked current data and a common situational awareness. Working with the FAA and airline community, Metron, Inc. and Volpe National Transportation Systems Center improved the process by utilizing real-time data exchange between all users, new algorithms to assign flight arrival slots, and new software in place at FAA facilities and airlines.**Fingerhut Companies, Inc., “Mail Stream Optimization”**Fingerhut mails up to 120 catalogs per year to each of its 6 million customers. With this dense mail plan and independent mailing decisions, many customers were receiving redundant and unproductive catalogs. To find and eliminate these unproductive catalogs, optimization models were developed to select the optimal sequence of catalogs, called a "mail stream", for each customer. With mail streams, Fingerhut is able to make mailing decisions at the customer level as well as increase profits. Today, this application runs weekly using current data to find the most profitable mail stream for each of its 6 million customers.**Ford, “Rightsizing and Management of Prototype Vehicle**Testing at Ford Motor Company” Prototype vehicles are used to verify new designs and represent a major annual investment at Ford Motor Company. Engineering managers studying in a Wayne State University master's degree program adapted a classroom set covering example and launched the development of the Prototype Optimization Model (POM). The POM is used for both operational and strategic planning. The modeling approach was lean and rapid and was designed to maintain the role of the experienced manager as the ultimate decision-maker.**IBM, “Matching Assets with Demand in Supply Chain**Management at IBM Microelectronics” The IBM Microelectronics Division is a leading-edge producer of semiconductor and packaged solutions supplying a wide range of customers inside and outside IBM. A critical component of customer responsiveness is matching assets with demand to correctly assess anticipated supplies linked with demand and provide manufacturing guidelines. A suite of tools was developed to handle matching in a division-wide "best can do", division wide ATP, and daily individual manufacturing location MRPs. The key modeling advance is the dynamic interweaving of linear programming, traditional MRP explosion and implosion-based heuristics and the ability to harness deep computing to solve large linear programming problems.**For More Information**• http://www.lionhrtpub.com/ORMS.shtml • http://www.informs.org/ • http://dsi.gsu.edu/**Application to InfoSec?**• Linear Programming • Queuing Theory/Simulation • Project management • Forecasting • Decision Analysis/Decision Trees**A Visual Model of the**Problem-Solving Process Identify Problem Formulate and Implement Model Analyze Model Test Results Implement Solution unsatisfactory results**Linear Programming**• Problems characterized by • limited resources. • decisions about how best to utilize the limited resources available to an individual or a business. • maximization or minimization of profits or costs. • Mathematical Programming (MP) is a field of management science that finds the optimal, or most efficient, way of using limited resources to achieve the objectives of an individual or a business. MP is sometimes referred to as optimization.**Optimization Example**Seuss’s Sandwich Shop sells two types of sandwiches: green eggs and ham (GEH) and ham and cheese (HC). A green eggs and ham sandwich consists of 2 slices bread, 1 green egg, and 1 slice ham. It takes an employee 3 minutes to make one of these sandwiches. A ham and cheese sandwich consists of 2 slices bread, 2 slices ham and 1 slice cheese. It takes 2 minutes to make a ham and cheese sandwich. The sandwich shop presently has available 400 slices of bread, 80 slices cheese, 120 green eggs and 200 slices of ham. The shop also has one employee scheduled for 7 hours to make all of the sandwiches. If a green egg and ham sandwich sells for $5 and a ham and cheese sandwich sells for $4, how many of each type should be prepared to maximize revenue? (Assume that demand is great enough to ensure that all sandwiches made will be sold).**Post-Optimality Analysis**• Range of Optimality – tells range that a decision variable’s coefficient can take on in the objective function without affecting current optimal solution (note that the objective function value WILL change). • Shadow/dual price – tells how receiving additional units of a resource affects the objective function value (for < constraints). Also tells how requiring more of something (for > constraints) affects the objective function value. Note that the changes in right hand side values are only good for a relevant range.**Other Applications in LP**1. (Operations planning) Diagnostic Corporation assembles two types of electronic calculators. The DC1 Calculator provides basic math functions, while the DC2 also provides trigonometric computations needed by engineers. Due to a winter blizzard, incoming shipments of components have been delayed. Ron Beckman, Manager of the plant, has assembled his production staff to plan an appropriate response. Bob Driscoll, in charge of supplies, reports that only three items are in short supply and likely to run out before new shipments arrive. The items in short supply are diodes (16,000 available), digital displays (10,000 available), and resistors (18,000 on hand). The quantities of each of these components that are required by each calculator are shown below. Mr. Beckman states that he would like to plan production to maximize the profits that can be realized using the available supply of parts. Number of Parts RequiredProfit Per ModelDiodesDisplaysResistorsCalculator DC1 6 2 6 $10 DC2 4 3 1 $12**Other Applications in LP**2. (Sales promotion) Riverside Auto wants to conduct an advertising campaign where each person who comes to the lot to look at a car receives $1. To advertise this campaign, Riverside can buy time on two local TV stations. The advertising agency has provided Riverside with the following data: Cost per Number of Number of Maximum Station Ad Serious Buyers Freeloaders Spots Ch. 47 $90 80 480 20 Ch. 59 $100 100 360 20 Riverside wants to minimize costs, while giving away a maximum of $15,000, and ensuring that the ads attract at least 2400 serious buyers. Formulate Riverside’s problem as a linear program.**Other Applications in LP**3. (Staff Scheduling) Palm General Hospital is concerned with the staffing of its emergency department. A recent analysis indicated that a typical day may be divided into six periods with the following requirements for nurses: Time Period Nurses Needed 7AM – 11AM 8 11AM – 3PM 6 3PM – 7PM 12 7PM – 11PM 6 11PM – 3AM 4 3AM – 7AM 2**Other Applications in LP**4. (Agriculture) A farming organization operates 3 farms of comparable productivity. The output of each farm is limited both by the usable acreage and by the amount of water available for irrigation. Data for the upcoming season are as follows: FarmUsable AcreageWater Available (Acre ft.) 1 400 1500 2 600 2000 3 300 900 The organization is considering 3 crops for planting, which differ in their expected profit/acre and in their consumption of water. Furthermore, the total acreage that can be devoted to each of the crops is limited by the amount of appropriate harvesting equipment available: Max Water Consumption Expected Profit CropAcreage in Acre ft./Acre Per Acre A 700 5 $400 B 800 4 $300 C 300 3 $100 In order to maintain a uniform workload among farms, the percentage of usable acreage planted must be the same at each farm. However, any combination of crops may be grown at any of the farms. Hw much of each crop should be planted at each farm to maximize expected profit?**Other Applications in LP**5. (Blending) Suppose that an oil refinery wishes to blend 4 petroleum constituents into 3 grades of gasoline A, B, and C. The availability and costs of the 4 constituents are as follows: ConstituentMax Availability (bbls/day)Cost per Barrel W 3,000 $3 X 2,000 $6 Y 4,000 $4 Z 1,000 $5 To maintain the required quality for each grade of gasoline it is necessary to maintain the following maximum and/or minimum percentages of the constituents in each blend. Determine the mix of the 4 constituents that will maximize profit. GradeSpecificationSelling Price per Barrel A no more than 30% of W $5.50 at least 40% of X no more than 50% of Y B no more than 50% of W $4.50 at least 10% of X C no more than 70% of W $3.50**Queuing Theory/Simulation**• Probabilistic Models – must make decision today, but don’t know for sure what will happen. • Americans are reported to spend 37 billion hours a year waiting in line! • E-mail often waits in a queue (i.e., line) on the Internet at intermediate computing centers before sent to final destination. • Subassemblies often wait in a line at machining centers to have the next operations performed. • Queuing theory represents the body of knowledge dealing with waiting lines.**Queuing Theory**• Conceived in the early 1900s when Danish telephone engineer, A.K. Erlang, began studying the congestion and waiting times occurring in the completion of telephone calls. • Since that time, a number of quantitative models have been developed to help business people understand waiting lines and make better decisions about how to manage them.**Queuing Theory**• Any time there is a finite capacity for service, you have a queuing system. • Channels and Stages: Channel: How many servers available for initial operation step? Stage: How many servers must an individual entity see before service is completed?**Queuing TheoryChannels and Stages**• Single Channel(# of servers available at each stage) / Single Stage(how many servers the entity must see before service is complete. Service Facility Input Source**Queuing TheoryChannels and Stages**• Multiple Channel(# of servers available at each stage) / Single Stage(how many servers the entity must see before service is complete. Input Source Service Facility Note: Every entity joins the same line and waits for the Next available server.**Queuing TheoryChannels and Stages**• Question: What is the difference between the following? • Multiple Channel/Single Stage • Multiple – Single Channel Single Stage Systems Input Source Service facility Service facility Input Source Service facility Input Source**Other FormsQueuing Systems**• Single Channel/Multiple Stage • Multiple Channel/Multiple Stage Input Source Input Source**Queuing Theory**• Managers use queuing theory to answer: • How many servers should we have? • How long should a customer wait, on average? • Queue Discipline: • Infinite Calling Population • Infinite Queue Capacity • No Balking or Reneging • First Come First Served • Requires that service rate is greater than or equal to arrival rate. • In multiple server case, all servers are of equal capability.**Queuing TheoryProcesses**• Input process can be either deterministic (D), general (G), or it can follow a Poisson Process (M). Poisson Process says that I know, on average, how many customers arrive per unit of time. Average arrival RATE is represented by l. • Service process can be either deterministic, general or follow a Poisson Process. Average service RATE is represented by m. • Number of servers is represented by K.**Queuing TheoryKendall’s Notation**• There are infinite numbers of possible queuing systems: 3 Possible Arrival Processes X 3 Possible Service Processes X Infinite Possible Number of Servers • To make sense of this, we use standard notation – Kendall’s Notation. Input Process Service Process # of Servers Queue Capacity M / M / 1 / ¥ M / M / 2 / ¥ M / M / K / ¥ M / G / K / ¥ G / M / K / ¥ G / G / K / ¥**Queuing TheoryM/M/1 Example**• Average Arrival Rate l=45 customers per hour • Average Service Rate m=60 customers per hour Questions of interest: Average time a customer spends waiting in line? Average number of customers waiting in line? Average time a customer spends in the system? Average number of customers in the system? Probability that the system is empty and idle?**Little’s Flow Equationsfor M/M/n/¥**NOTE: These are same as for M/M/1/¥**Values for P0 for Multiple Channel Waiting Lines with**Poisson Arrivalsand Exponential Service Times P0**Queuing TheoryM/M/2 Example**• Average Arrival Rate l=45 customers per hour • Average Service Rate m=60 customers per hour • Number of Channels, K, = 2 NOTE: l/m=45/60=0.75 – this value is on lookup table for k=2 0.4545**Cost/Service Tradeoffs**To Recap Service Statistics: So, is it worth the extra cost to add an additional server? Suppose servers earn $15 per hour, but customer cost of waiting (loss of goodwill, etc.) has been estimated to be $25 per hour. **Cost/Service Tradeoffs (cont’d)**• Customer waiting cost can either be associated with queue time or total time in system. We’ll go with “total time in system.” • Therefore, our per-hour total (system) cost is represented by TC=(Cs X K)+ (CW X L) where Cs = hourly cost for server and CW = hourly waiting cost TCk=1 =(15 X 1) + (25 X 3) = $90 TCk=2 =(15 X 2) + (25 X 0.8730) = $51.83**Simulation**• Assumptions of Queuing - Revisited • Infinite Calling Population • Infinite Queue Capacity • No Balking or Reneging • First Come First Served • Requires that service rate is greater than or equal to arrival rate. • In multiple server case, all servers are of equal capability. • Note that the above assumptions are fairly restrictive! Simulation offers us flexibility that Queuing Theory does not.**Simulation**• Simulation is a quick way to model long periods of time. • Simulation requires that we generate a stream of numbers that are random and that have no relationship to each other. The RANDOM NUMBER GENERATOR IS KEY. • In MS Excel, I can use =rand() to generate a random number greater than 0.00 but less than 1.00. • Can humans generate random numbers?**Simulation Example**0.10 no purchase 0.75 no answer 0.30 1 0.40 Sales- person House 2 0.30 Purchase 0.25 3 0.20 M 0.20 4 answer 0.70 0.60 1 2 purchase 0.15 0.30 F 0.80 3 0.10 no purchase 0.85**Simulation Example (cont’d)**We now need to construct probability distributions associated with each event in our simulation example Answer?ProbabilityRNUMGenderProbabilityRNUM Y 0.7 0.00-0.69 M 0.2 0.00-0.19 N 0.3 0.70-0.99 F 0.8 0.20-0.99 Female Sale?ProbabilityRNUMMale Sale?ProbabilityRNUM Y 0.15 0.00-0.14 Y 0.25 0.00-0.24 N 0.85 0.15-0.99 N 0.75 0.25-0.99 Number (F)?ProbabilityRNUMNumber (M)?ProbabilityRNUM 1 0.60 0.00-0.59 1 0.10 0.00-0.09 2 0.30 0.60-0.89 2 0.40 0.10-0.49 3 0.10 0.90-0.99 3 0.30 0.50-0.79 4 0.20 0.80-0.99**Forecasting**• A forecast is an estimate of future demand • Forecasts contain error • Forecasts can be created by subjective means by estimates from informal sources • OR forecasts can be determined mathematically by using historical data • OR forecasts can be based on both subjective and mathematical techniques.

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