2Chapter 12 Advanced Optimization Applications Data Envelopment AnalysisRevenue ManagementPortfolio Models and Asset AllocationNonlinear OptimizationConstructing an Index Fund
3Data Envelopment Analysis Data envelopment analysis (DEA) is an LP application used to determine the relative operating efficiency of units with the same goals and objectives.DEA creates a fictitious composite unit made up of an optimal weighted average (W1, W2,…) of existing units.An individual unit, k, can be compared by determining E, the fraction of unit k’s input resources required by the optimal composite unit.If E < 1, unit k is less efficient than the composite unit and be deemed relatively inefficient.If E = 1, there is no evidence that unit k is inefficient, but one cannot conclude that k is absolutely efficient.
4Data Envelopment Analysis The DEA ModelMin Es.t. Weighted outputs > Unit k’s output(for each measured output)Weighted inputs < E [Unit k’s input](for each measured input)Sum of weights = 1E, weights > 0
5Data Envelopment Analysis The Langley County School District is trying to determine the relative efficiency of its three high schools. In particular, it wants to evaluate Roosevelt High. The district is evaluating performances on SAT scores, the number of seniors finishing high school, and the number of students who enter college as a function of the number of teachers teaching senior classes, the prorated budget for senior instruction, and the number of students in the senior class.
6Data Envelopment Analysis InputRoosevelt Lincoln WashingtonSenior FacultyBudget ($100,000's)Senior Enrollments
7Data Envelopment Analysis OutputRoosevelt Lincoln WashingtonAverage SAT ScoreHigh School GraduatesCollege Admissions
8Data Envelopment Analysis Define the Decision VariablesE = Fraction of Roosevelt's input resources required by the composite high schoolw1 = Weight applied to Roosevelt's input/output resources by the composite high schoolw2 = Weight applied to Lincoln’s input/output resources by the composite high schoolw3 = Weight applied to Washington's input/output resources by the composite high school
9Data Envelopment Analysis Define the Objective FunctionMinimize the fraction of Roosevelt High School's input resources required by the composite high school:Min E
10Data Envelopment Analysis Define the ConstraintsSum of the Weights is 1:(1) w1 + w2 + w3 = 1Output Constraints:Since w1 = 1 is possible, each output of the composite school must be at least as great as that of Roosevelt:(2) 800w w w3 > (SAT Scores)(3) 450w w w3 > (Graduates)(4) 140w w w3 > (College Admissions)
11Data Envelopment Analysis Define the Constraints (continued)Input Constraints:The input resources available to the composite school is a fractional multiple, E, of the resources available to Roosevelt. Since the composite high school cannot use more input than that available to it, the input constraints are:(5) 37w w w3 < 37E (Faculty)(6) 6.4w w w3 < 6.4E (Budget)(7) 850w w w3 < 850E (Seniors)Nonnegativity of variables:E, w1, w2, w3 > 0
12Data Envelopment Analysis Computer SolutionObjective Function Value =Variable Value Reduced CostEWWW
13Data Envelopment Analysis ConclusionThe output shows that the composite school is made up of equal weights of Lincoln and Washington. Roosevelt is 76.5% efficient compared to this composite school when measured by college admissions (because of the 0 slack on this constraint #4). It is less than 76.5% efficient when using measures of SAT scores and high school graduates (there is positive slack in constraints 2 and 3.)
14Revenue Management Another LP application is revenue management. Revenue management involves managing the short-term demand for a fixed perishable inventory in order to maximize revenue potential.The methodology was first used to determine how many airline seats to sell at an early-reservation discount fare and many to sell at a full fare.Application areas now include hotels, apartment rentals, car rentals, cruise lines, and golf courses.
15Revenue Management LeapFrog Airways provides passenger service for Indianapolis, Baltimore, Memphis, Austin, and Tampa.LeapFrog has two WB828 airplanes, one based in Indianapolis and the other in Baltimore. Each morningthe Indianapolis based plane flies to Austin with a stopover in Memphis. The Baltimore based plane flies toTampa with a stopover in Memphis. Both planes have a coach section with a 120-seat capacity.
16Revenue Management LeapFrog uses two fare classes: a discount fare D class and a full fare F class. Leapfrog’s products, eachreferred to as an origin destination itinerary fare (ODIF), are listed on the next slide with their fares andforecasted demand.LeapFrog wants to determine how many seats it should allocate to each ODIF.
17Revenue Management Fare Class D F ODIF Code IMD IAD ITD IMF IAF ITF BMDBADBTDBMFBAFBTFMADMTDMAFMTFODIF12345678910111213141516OriginIndianapolisBaltimoreMemphisDestinationMemphisAustinTampaTampa AustinFare175275285395425475185315290385525490190180310295Demand442540151082650421216958481411
18Revenue Management Define the Decision Variables There are 16 variables, one for each ODIF:IMD = number of seats allocated to Indianapolis-Memphis-Discount classIAD = number of seats allocated to Indianapolis-Austin- Discount classITD = number of seats allocated to Indianapolis-Tampa- Discount classIMF = number of seats allocated to Indianapolis-Memphis- Full Fare classIAF = number of seats allocated to Indianapolis-Austin-Full Fare class
19Revenue Management Define the Decision Variables (continued) ITF = number of seats allocated to Indianapolis-Tampa-Full Fare classBMD = number of seats allocated to Baltimore-Memphis-Discount classBAD = number of seats allocated to Baltimore-Austin-BTD = number of seats allocated to Baltimore-Tampa-BMF = number of seats allocated to Baltimore-Memphis-BAF = number of seats allocated to Baltimore-Austin-
20Revenue Management Define the Decision Variables (continued) BTF = number of seats allocated to Baltimore-Tampa-Full Fare classMAD = number of seats allocated to Memphis-Austin-Discount classMTD = number of seats allocated to Memphis-Tampa-MAF = number of seats allocated to Memphis-Austin-MTF = number of seats allocated to Memphis-Tampa-
21Revenue Management Define the Objective Function Maximize total revenue:Max (fare per seat for each ODIF)x (number of seats allocated to the ODIF)Max 175IMD + 275IAD + 285ITD + 395IMF+ 425IAF + 475ITF + 185BMD + 315BAD+ 290BTD + 385BMF + 525BAF + 490BTF+ 190MAD + 180MTD + 310MAF + 295MTF
22Revenue Management Define the Constraints Indianapolis-Memphis leg There are 4 capacity constraints, one for each flight leg:Indianapolis-Memphis leg(1) IMD + IAD + ITD + IMF + IAF + ITF < 120Baltimore-Memphis leg(2) BMD + BAD + BTD + BMF + BAF + BTF < 120Memphis-Austin leg(3) IAD + IAF + BAD + BAF + MAD + MAF < 120Memphis-Tampa leg(4) ITD + ITF + BTD + BTF + MTD + MTF < 120
23Revenue Management Define the Constraints (continued) There are 16 demand constraints, one for each ODIF:(5) IMD < 44 (11) BMD < 26 (17) MAD < 5(6) IAD < 25 (12) BAD < 50 (18) MTD < 48(7) ITD < 40 (13) BTD < 42 (19) MAF < 14(8) IMF < 15 (14) BMF < 12 (20) MTF < 11(9) IAF < 10 (15) BAF < 16(10) ITF < 8 (16) BTF < 9
24Revenue Management Computer Solution Objective Function Value =Variable Value Reduced CostIMDIADITDIMFIAFITFBMDBAD
25Portfolio Models and Asset Management Asset allocation involves determining how to allocate investment funds across a variety of asset classes such as stocks, bonds, mutual funds, real estate.Portfolio models are used to determine percentage of funds that should be made in each asset class.The goal is to create a portfolio that provides the best balance between risk and return.
26Portfolio Model John Sweeney is an investment advisor who is attempting to construct an "optimal portfolio" for aclient who has $400,000 cash to invest. There are tendifferent investments, falling into four broad categories that John and his client have identified as potential candidate for this portfolio.The investments and their important characteristics are listed in the table on the next slide. Note that Unidyde Corp. under Equities and Unidyde Corp. under Debt are two separate investments, whereas First General REIT is a single investment that is considered both an equities and a real estate investment.
27Portfolio Model Exp. Annual After Tax Liquidity Risk Category Investment Return Factor FactorEquities Unidyde Corp %(Stocks) CC’s Restaurants %First General REIT %Debt Metropolis Electric %(Bonds) Unidyde Corp %Lewisville Transit %Real Estate Realty Partners 22.0%First General REIT ( --- See above --- )Money T-Bill Account %Money Mkt. Fund %Saver's Certificate %
28Portfolio Model Formulate a linear programming problem to accomplish John's objective as an investment advisorwhich is to construct a portfolio that maximizes hisclient's total expected after-tax return over the next year, subject to the limitations placed upon him by the client for the portfolio. (Limitations listed on next two slides.)
29Portfolio ModelPortfolio Limitations1. The weighted average liquidity factor for the portfoliomust to be at least 65.2. The weighted average risk factor for the portfolio mustbe no greater than 55.3. No more than $60,000 is to be invested in Unidydestocks or bonds.4. No more than 40% of the investment can be in any onecategory except the money category.5. No more than 20% of the total investment can be inany one investment except the money market fund.continued
30Portfolio Model Portfolio Limitations (continued) 6. At least $1,000 must be invested in the Money Marketfund.7. The maximum investment in Saver's Certificates is$15,000.8. The minimum investment desired for debt is $90,000.9. At least $10,000 must be placed in a T-Bill account.
31Portfolio Model Define the Decision Variables X1 = $ amount invested in Unidyde Corp. (Equities)X2 = $ amount invested in CC’s RestaurantsX3 = $ amount invested in First General REITX4 = $ amount invested in Metropolis ElectricX5 = $ amount invested in Unidyde Corp. (Debt)X6 = $ amount invested in Lewisville TransitX7 = $ amount invested in Realty PartnersX8 = $ amount invested in T-Bill AccountX9 = $ amount invested in Money Mkt. FundX10 = $ amount invested in Saver's Certificate
32Portfolio Model Define the Objective Function Maximize the total expected after-tax return over the next year:Max .15X X X X X5+ .12X X X X X10
33Portfolio Model Define the Constraints Total funds invested must not exceed $400,000:(1) X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 = 400,000Weighted average liquidity factor must to be at least 65:100X X X3 + 95X4 + 92X5 + 79X6 + 80X X9 >65(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)Weighted average risk factor must be no greater than 55:60X1 + 70X2 + 75X3 + 20X4 + 30X5 + 22X6 + 50X7 + 10X9 <55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)No more than $60,000 to be invested in Unidyde Corp:X1 + X5 < 60,000
34Portfolio Model Define the Constraints (continued) No more than 40% of the $400,000 investment can bein any one category except the money category:(5) X1 + X2 + X3 < 160,000(6) X4 + X5 + X6 < 160,000X3 + X7 < 160,000No more than 20% of the $400,000 investment can bein any one investment except the money market fund:(8) X2 < 80,000 (12) X7 < 80,000(9) X3 < 80,000 (13) X8 < 80,000(10) X4 < 80,000 (14) X10 < 80,000(11) X6 < 80,000
35Portfolio Model Define the Constraints (continued) At least $1,000 must be invested in the Money Market fund:(15) X9 > 1,000The maximum investment in Saver's Certificates is $15,000:(16) X10 < 15,000The minimum investment the Debt category is $90,000:(17) X4 + X5 + X6 > 90,000At least $10,000 must be placed in a T-Bill account:(18) X8 > 10,000Non-negativity of variables:Xj > j = 1, , 10
36Portfolio Model Solution Summary Total Expected After-Tax Return = $64,355X1 = $ invested in Unidyde Corp. (Equities)X2 = $80,000 invested in CC’s RestaurantsX3 = $80,000 invested in First General REITX4 = $ invested in Metropolis ElectricX5 = $60,000 invested in Unidyde Corp. (Debt)X6 = $74,000 invested in Lewisville TransitX7 = $80,000 invested in Realty PartnersX8 = $10,000 invested in T-Bill AccountX9 = $1,000 invested in Money Mkt. FundX10 = $15,000 invested in Saver's Certificate
37Conservative Portfolio A portfolio manager has been asked to develop a portfolio for the firm’s conservative clients who express a strong aversion to risk. The manager’s task is to determine the proportion of the portfolio to invest in each of six mutual funds so that the portfolio provides the best return possible with a minimum risk.The annual return (%) for five 1-year periods for the six mutual funds are shown on the next slide. The portfolio manager thinks that the returns for the five years shown in the table are scenarios that can be used to represent the possibilities for the next year.
39Conservative Portfolio Define the Decision VariablesFS = proportion invested in foreign stock mutual fundIB = proportion invested in intermediate-term bond fundLG = proportion invested in large-cap growth fundLV = proportion invested in large-cap value fundSG = proportion invested in small-cap growth fundSV = proportion invested in small-cap value fund
40Conservative Portfolio ConstraintsMinimum returns for five scenarios:– M FS IB LG LV SG SV ≥ 0– M FS IB LG LV SG SV ≥ 0– M FS IB LG LV SG – SV ≥ 0– M FS – IB LG LV SG SV ≥ 0– M – 21.93FS IB – 23.26LG – LV – SG SV ≥ 0Sum of the proportions must equal 1:FS + IB + LG + LV + SG + SV = 1Non-negativityM, FS, IB, LG, LV, SG, SV ≥ 0Objective FunctionMaximize the minimum return for the portfolio:Max M
41Conservative Portfolio Optimal SolutionThe optimal value of the objective function is(The optimal portfolio will earn 6.445% in the worst-case scenario.)55.4% of the portfolio should be invested in theintermediate-term bond fund.13.2% of the portfolio should be invested in thelarge-cap growth fund.31.4% of the portfolio should be invested in thesmall-cap value fund.
42Moderate PortfolioA portfolio manager would like to construct a portfolio for clients who are willing to accept a moderate amount of risk in order to attempt to achieve better returns. Suppose that clients in this risk category are willing to accept some risk, but do not want the annual return for the portfolio to drop below 2%.The annual return (%) for five 1-year periods for the six mutual funds are shown on the next slide. The portfolio manager thinks that the returns for the five years shown in the table are scenarios that can be used to represent the possibilities for the next year.
43Moderate Portfolio Constraints Minimum returns for five scenarios: – M FS IB LG LV SG SV ≥ 2– M FS IB LG LV SG SV ≥ 2– M FS IB LG LV SG – SV ≥ 2– M FS – IB LG LV SG SV ≥ 2– M – 21.93FS IB – 23.26LG – LV – SG SV ≥ 2Sum of the proportions must equal 1:FS + IB + LG + LV + SG + SV = 1Non-negativityM, FS, IB, LG, LV, SG, SV ≥ 0
44Moderate Portfolio Objective Function The coefficient of FS in the objective function is given by:0.2(10.06) + 0.2(13.12) + 0.2(13.47) + 0.2(45.42) + 0.2( – 21.93)The coefficient of IB is given by:0.2(17.64) + 0.2(3.25) + 0.2(7.51) = 0.2( – 1.33) = 0.2(7.36) = 6.89… and so on. Thus, the objective function is:Maximize the minimum return for the portfolio:Max 12.03FS IB LG LV SG SV
45Moderate Portfolio Optimal Solution Invest 10.8% of the portfolio in a large-cap growthmutual fund.Invest 41.5% in a small-cap growth mutual fund.Invest 47.7% in a small-cap value mutual fund.This allocation provides a maximum expected return of 17.33%.The portfolio return will only be 2% if scenarios 3 or 5 occur (constraints 3 and 5 are binding).The portfolio return will be % if scenario 1 occurs, % if scenario 2 occurs, and % if scenario 4 occurs.
46Nonlinear Optimization Many business processes behave in a nonlinear manner.The price of a bond is a nonlinear function of interest rates.The price of a stock option is a nonlinear function of the price of the underlying stock.The marginal cost of production often decreases with the quantity produced.The quantity demanded for a product is often a nonlinear function of the price.
47Nonlinear Optimization A nonlinear optimization problem is any optimization problem in which at least one term in the objective function or a constraint is nonlinear.Nonlinear terms includeThe nonlinear optimization problems presented on the upcoming slides can be solved using computer software such as LINGO and Excel Solver.
48Nonlinear Optimization Armstrong Bike Co.Armstrong Bike Co. produces two new lightweightbicycle frames, the Flyer and the Razor, that are madefrom special aluminum and steel alloys. The cost toproduce a Flyer frame is $100, and the cost to produce aRazor frame is $120.We can not assume that Armstrong will sell all theframes it can produce. As the selling price of each framemodel – Flyer and Razor - increases, the quantitydemanded for each model goes down.
49Nonlinear Optimization Assume that the demand for Flyer frames Fand the demand for Razor frames R are given by:F = 750 – 5PFR = 400 – 2PRwhere PF = the price of a Flyer framePR = the price of a Razor frame.The profit contributions (revenue – cost) are:PF F - 100F for Flyer framesPR R - 120R for Razor frames
50Nonlinear Optimization Profit Contribution as a Function of DemandSolving F = PF for PF we get:PF = /5 FSubstituting /5 F for PF in PF F - 100F we get:PF F - 100F = F( /5 F) - 100F = 50F - 1/5 F 2Solving R = PR for PR we get:PR = /2 RSubstituting /2 R for PR in PR R - 120R we get:PR R - 120R = R( /2 R) - 120R = 80R - 1/2 R2
51Nonlinear Optimization Total Profit ContributionTotal Profit Contribution = 50F – 1/5 F2 + 80R – 1/2 R2This function is an example of a quadratic functionbecause the nonlinear terms have a power of 2.
52Nonlinear Optimization A supplier can deliver a maximum of 500pounds of the aluminum alloy and 420 pounds of thesteel alloy weekly. The number of pounds of each alloyneeded per frame is summarized below.Aluminum Alloy Steel AlloyFlyerRazorHow many Flyer and Razor frames shouldArmstrong produce each week?
53Nonlinear Optimization Problem FormulationMax 50F – 1/5 F2 + 80R – 1/2 R2 (Total Weekly Profit)s.t F + 4R < (Aluminum Available)3F + 2R < (Steel Available)F, R > (Non-negativity)
54Nonlinear Optimization Total Profit ContributionFirst, we will solve the unconstrained version ofthis nonlinear program to find the values of F and Rthat maximize the above total profit contributionfunction (with the production constraints ignored).
55An Unconstrained Problem Optimal Solution for Unconstrained Problemx2250200150100503F + 2R < 420UnconstrainedOptimum(125, 80)Profit = $6,325.00FeasibleRegion2F + 4R < 500x1
56A Constrained Problem Total Profit Contribution Now we will solve the constrained version of thisnonlinear program to find the values of F and R thatmaximize the total profit contribution function withthe production constraints enforced.
57A Constrained Problem Objective Function Contour Lines x2 x1 250 200 15010050$6,200.00Contour$6,325.00$5,500.00Contour$6,075.47Contourx1
58A Constrained Problem Optimal Solution for Constrained Problem x2 x1 25020015010050ConstrainedOptimum(92.45, 71.32)Profit = $6,075.47$6,075.47Contourx1
59A Constrained Problem Optimal Solution Produce Flyer frames per week.Produce Razor frames per week.Profit per week is $6,Use pounds of aluminum alloy per week (of the 500 pounds available per week).Use the entire 420 pounds of steel alloy available per week.
60Local and Global Optima A feasible solution is a local optimum if there are no other feasible solutions with a better objective function value in the immediate neighborhood.For a maximization problem the local optimum corresponds to a local maximum.For a minimization problem the local optimum corresponds to a local minimum.A feasible solution is a global optimum if there are no other feasible points with a better objective function value in the feasible region.Obviously, a global optimum is also a local optimum.
61Multiple Local OptimaNonlinear optimization problems can have multiple local optimal solutions, in which case we want to find the best local optimum.Nonlinear problems with multiple local optima are difficult to solve and pose a serious challenge for optimization software.In these cases, the software can get “stuck” and terminate at a local optimum.There can be a severe penalty for finding a local optimum that is not a global optimum.Developing algorithms capable of finding the global optimum is currently a very active research area.
62Multiple Local Optima Consider the function The shape of this function is shown on the next slide.The hills and valleys in the graph show that this function has several local maximums and local minimums.There are two local minimums, one of which is the the global minimum.There are three local maximums, one of which is the global maximum.
64Single Local Optimum Consider the function The shape of this function is shown on the next slide.A function that is bowl-shaped down is called a concave function.The maximum value for this particular function is 0 and the point (0, 0) gives the optimal value of 0.Functions such as this one have a single local maximum that is also a global maximum.This type of nonlinear problem is relatively easy to maximize.
66Single Local Optimum Consider the function The shape of this function is shown on the next slide.A function that is bowl-shaped up is called a convex function.The minimum value for this particular function is 0 and the point (0, 0) gives the optimal value of 0.Functions such as this one have a single local minimum that is also a global minimum.This type of nonlinear problem is relatively easy to minimize.
68Constructing an Index Fund Index funds are a very popular investment vehicle in the mutual fund industry.Vanguard 500 Index Fund is the largest mutual fund in the U.S. with over $70 billion in net assets in 2005.An index fund is an example of passive asset management.The key idea behind an index fund is to construct a portfolio of stocks, mutual funds, or other securities that closely matches the performance of a broad market index such as the S&P 500.Behind the popularity of index funds is research that basically says “you can’t beat the market.”
69Example: Constructing an Index Fund Lymann Brothers InvestmentsLymann Brothers has a substantial number of clientswho wish to own a mutual fund portfolio that closelymatches the performance of the S&P 500 stock index.A manager at Lymann Brothers has selected fivemutual funds (shown on the next slide) that will beconsidered for inclusion in the portfolio. The managermust decide what percentage of the portfolio should beinvested in each mutual fund.
70Example: Constructing an Index Fund Mutual Fund Performance in 4 Selected YearsAnnual Returns (Planning Scenarios)Mutual Fund Year 1 Year 2 Year 3 Year 4International StockLarge-Cap BlendMid-Cap BlendSmall-Cap BlendIntermediate BondS&P
71Example: Constructing an Index Fund Define the 9 Decision VariablesIS = proportion of portfolio invested in international stockLC = proportion of portfolio invested in large-cap blendMC = proportion of portfolio invested in mid-cap blendSC = proportion of portfolio invested in small-cap blendIB = proportion of portfolio invested in intermediate bondR1 = portfolio return for scenario 1 (year 1)R2 = portfolio return for scenario 2 (year 2)R3 = portfolio return for scenario 3 (year 3)R4 = portfolio return for scenario 4 (year 4)
72Example: Constructing an Index Fund Define the Objective FunctionMin (R1 – 13)2 + (R2 – 12)2 + (R3 – 7)2 + (R4 – 2)2Define the 6 Constraints (including non-negativity)25.64IS LC MC SC IB = R127.62IS LC MC SC IB = R25.80IS LC MC SC IB = R3- 3.13IS LC MC SC IB = R4IS + LC + MC + SC + IB = 1IS, LC, MC, SC, IB > 0
73Example: Constructing an Index Fund Optimal Solution for Lymann Brothers ExampleR1 = (12.51% portfolio return for scenario 1)R2 = (12.90% portfolio return for scenario 2)R3 = ( 7.13% portfolio return for scenario 3)R4 = ( 2.51% portfolio return for scenario 4)IS = ( 0.0% of portfolio in international stock)LC = ( 0.0% of portfolio in large-cap blend)MC = (33.2% of portfolio in mid-cap blend)SC = (16.1% of portfolio in small-cap blend)IB = (50.7% of portfolio in intermediate bond)100.0% of portfolio
74Example: Constructing an Index Fund Lymann Brothers Portfolio Return vs. S&P 500 ReturnScenario Portfolio Return S&P 500 Return