Slides by . John Loucks St. Edward’s Univ..

Slides by . John Loucks St. Edward’s Univ.

Chapter 12 Advanced Optimization Applications
Data Envelopment Analysis Revenue Management Portfolio Models and Asset Allocation Nonlinear Optimization Constructing an Index Fund

Data 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.

The DEA Model Min E s.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 = 1 E, weights > 0

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.

Input Roosevelt Lincoln Washington Senior Faculty Budget ($100,000's) Senior Enrollments

Output Roosevelt Lincoln Washington Average SAT Score High School Graduates College Admissions

Define the Decision Variables E = Fraction of Roosevelt's input resources required by the composite high school w1 = Weight applied to Roosevelt's input/output resources by the composite high school w2 = Weight applied to Lincoln’s input/output resources by the composite high school w3 = Weight applied to Washington's input/output resources by the composite high school

Define the Objective Function Minimize the fraction of Roosevelt High School's input resources required by the composite high school: Min E

Define the Constraints Sum of the Weights is 1: (1) w1 + w2 + w3 = 1 Output 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)

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

Computer Solution Objective Function Value = Variable Value Reduced Cost E W W W

Conclusion The 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.)

Revenue 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.

Revenue 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 morning the Indianapolis based plane flies to Austin with a stopover in Memphis. The Baltimore based plane flies to Tampa with a stopover in Memphis. Both planes have a coach section with a 120-seat capacity.

Revenue Management LeapFrog uses two fare classes: a discount fare D
class and a full fare F class. Leapfrog’s products, each referred to as an origin destination itinerary fare (ODIF), are listed on the next slide with their fares and forecasted demand. LeapFrog wants to determine how many seats it should allocate to each ODIF.

Revenue Management Fare Class D F ODIF Code IMD IAD ITD IMF IAF ITF
BMD BAD BTD BMF BAF BTF MAD MTD MAF MTF ODIF 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Origin Indianapolis Baltimore Memphis Destination Memphis Austin Tampa Tampa Austin Fare 175 275 285 395 425 475 185 315 290 385 525 490 190 180 310 295 Demand 44 25 40 15 10 8 26 50 42 12 16 9 58 48 14 11

Revenue Management Define the Decision Variables
There are 16 variables, one for each ODIF: IMD = number of seats allocated to Indianapolis-Memphis- Discount class IAD = number of seats allocated to Indianapolis-Austin- Discount class ITD = number of seats allocated to Indianapolis-Tampa- Discount class IMF = number of seats allocated to Indianapolis-Memphis- Full Fare class IAF = number of seats allocated to Indianapolis-Austin-Full Fare class

Revenue Management Define the Decision Variables (continued)
ITF = number of seats allocated to Indianapolis-Tampa- Full Fare class BMD = number of seats allocated to Baltimore-Memphis- Discount class BAD = 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-

Revenue Management Define the Decision Variables (continued)
BTF = number of seats allocated to Baltimore-Tampa- Full Fare class MAD = number of seats allocated to Memphis-Austin- Discount class MTD = number of seats allocated to Memphis-Tampa- MAF = number of seats allocated to Memphis-Austin- MTF = number of seats allocated to Memphis-Tampa-

Revenue 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

Revenue 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 < 120 Baltimore-Memphis leg (2) BMD + BAD + BTD + BMF + BAF + BTF < 120 Memphis-Austin leg (3) IAD + IAF + BAD + BAF + MAD + MAF < 120 Memphis-Tampa leg (4) ITD + ITF + BTD + BTF + MTD + MTF < 120

Revenue 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

Revenue Management Computer Solution
Objective Function Value = Variable Value Reduced Cost IMD IAD ITD IMF IAF ITF BMD BAD

Portfolio 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.

Portfolio Model John Sweeney is an investment advisor who is
attempting to construct an "optimal portfolio" for a client who has $400,000 cash to invest. There are ten different 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.

Portfolio Model Exp. Annual After Tax Liquidity Risk
Category Investment Return Factor Factor Equities 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 %

Portfolio Model Formulate a linear programming problem to
accomplish John's objective as an investment advisor which is to construct a portfolio that maximizes his client'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.)

Portfolio Model Portfolio Limitations 1. The weighted average liquidity factor for the portfolio must to be at least 65. 2. The weighted average risk factor for the portfolio must be no greater than 55. 3. No more than $60,000 is to be invested in Unidyde stocks or bonds. 4. No more than 40% of the investment can be in any one category except the money category. 5. No more than 20% of the total investment can be in any one investment except the money market fund. continued

Portfolio Model Portfolio Limitations (continued)
6. At least $1,000 must be invested in the Money Market fund. 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.

Portfolio Model Define the Decision Variables
X1 = $ amount invested in Unidyde Corp. (Equities) X2 = $ amount invested in CC’s Restaurants X3 = $ amount invested in First General REIT X4 = $ amount invested in Metropolis Electric X5 = $ amount invested in Unidyde Corp. (Debt) X6 = $ amount invested in Lewisville Transit X7 = $ amount invested in Realty Partners X8 = $ amount invested in T-Bill Account X9 = $ amount invested in Money Mkt. Fund X10 = $ amount invested in Saver's Certificate

Portfolio 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

Portfolio Model Define the Constraints
Total funds invested must not exceed $400,000: (1) X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 = 400,000 Weighted 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

Portfolio Model Define the Constraints (continued)
No more than 40% of the $400,000 investment can be in any one category except the money category: (5) X1 + X2 + X3 < 160,000 (6) X4 + X5 + X6 < 160,000 X3 + X7 < 160,000 No more than 20% of the $400,000 investment can be in 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

Portfolio Model Define the Constraints (continued)
At least $1,000 must be invested in the Money Market fund: (15) X9 > 1,000 The maximum investment in Saver's Certificates is $15,000: (16) X10 < 15,000 The minimum investment the Debt category is $90,000: (17) X4 + X5 + X6 > 90,000 At least $10,000 must be placed in a T-Bill account: (18) X8 > 10,000 Non-negativity of variables: Xj > j = 1, , 10

Portfolio Model Solution Summary
Total Expected After-Tax Return = $64,355 X1 = $ invested in Unidyde Corp. (Equities) X2 = $80,000 invested in CC’s Restaurants X3 = $80,000 invested in First General REIT X4 = $ invested in Metropolis Electric X5 = $60,000 invested in Unidyde Corp. (Debt) X6 = $74,000 invested in Lewisville Transit X7 = $80,000 invested in Realty Partners X8 = $10,000 invested in T-Bill Account X9 = $1,000 invested in Money Mkt. Fund X10 = $15,000 invested in Saver's Certificate

Conservative 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.

Planning Scenarios Mutual Fund Year 1 Year 2 Year 3 Year 4 Year 5 Foreign Stock 10.06 13.12 13.47 45.42 -21.93 Intermediate-Term Bond 17.64 3.25 7.51 -1.33 7.36 Large-Cap Growth 32.41 18.71 33.28 41.46 -23.26 Large-Cap Value 32.36 20.61 12.93 7.06 -5.37 Small-Cap Growth 33.44 19.40 3.85 58.68 -9.02 Small-Cap Value 24.56 25.32 -6.70 5.43 17.31 S&P 500 Return 25.00 20.00 8.00 30.00 -10.00

Define the Decision Variables FS = proportion invested in foreign stock mutual fund IB = proportion invested in intermediate-term bond fund LG = proportion invested in large-cap growth fund LV = proportion invested in large-cap value fund SG = proportion invested in small-cap growth fund SV = proportion invested in small-cap value fund

Constraints Minimum 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 ≥ 0 Sum of the proportions must equal 1: FS + IB + LG + LV + SG + SV = 1 Non-negativity M, FS, IB, LG, LV, SG, SV ≥ 0 Objective Function Maximize the minimum return for the portfolio: Max M

Optimal Solution The 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 the intermediate-term bond fund. 13.2% of the portfolio should be invested in the large-cap growth fund. 31.4% of the portfolio should be invested in the small-cap value fund.

Moderate Portfolio A 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.

Moderate 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 ≥ 2 Sum of the proportions must equal 1: FS + IB + LG + LV + SG + SV = 1 Non-negativity M, FS, IB, LG, LV, SG, SV ≥ 0

Moderate 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

Moderate Portfolio Optimal Solution
Invest 10.8% of the portfolio in a large-cap growth mutual 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.

Nonlinear 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.

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 include The nonlinear optimization problems presented on the upcoming slides can be solved using computer software such as LINGO and Excel Solver.

Armstrong Bike Co. Armstrong Bike Co. produces two new lightweight bicycle frames, the Flyer and the Razor, that are made from special aluminum and steel alloys. The cost to produce a Flyer frame is $100, and the cost to produce a Razor frame is $120. We can not assume that Armstrong will sell all the frames it can produce. As the selling price of each frame model – Flyer and Razor - increases, the quantity demanded for each model goes down.

Assume that the demand for Flyer frames F and the demand for Razor frames R are given by: F = 750 – 5PF R = 400 – 2PR where PF = the price of a Flyer frame PR = the price of a Razor frame. The profit contributions (revenue – cost) are: PF F - 100F for Flyer frames PR R - 120R for Razor frames

Profit Contribution as a Function of Demand Solving F = PF for PF we get: PF = /5 F Substituting /5 F for PF in PF F - 100F we get: PF F - 100F = F( /5 F) - 100F = 50F - 1/5 F 2 Solving R = PR for PR we get: PR = /2 R Substituting /2 R for PR in PR R - 120R we get: PR R - 120R = R( /2 R) - 120R = 80R - 1/2 R2

Total Profit Contribution Total Profit Contribution = 50F – 1/5 F2 + 80R – 1/2 R2 This function is an example of a quadratic function because the nonlinear terms have a power of 2.

A supplier can deliver a maximum of 500 pounds of the aluminum alloy and 420 pounds of the steel alloy weekly. The number of pounds of each alloy needed per frame is summarized below. Aluminum Alloy Steel Alloy Flyer Razor How many Flyer and Razor frames should Armstrong produce each week?

Problem Formulation Max 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)

Total Profit Contribution First, we will solve the unconstrained version of this nonlinear program to find the values of F and R that maximize the above total profit contribution function (with the production constraints ignored).

An Unconstrained Problem
Optimal Solution for Unconstrained Problem x2 250 200 150 100 50 3F + 2R < 420 Unconstrained Optimum (125, 80) Profit = $6,325.00 Feasible Region 2F + 4R < 500 x1

A Constrained Problem Total Profit Contribution
Now we will solve the constrained version of this nonlinear program to find the values of F and R that maximize the total profit contribution function with the production constraints enforced.

A Constrained Problem Objective Function Contour Lines x2 x1 250 200
150 100 50 $6,200.00 Contour $6,325.00 $5,500.00 Contour $6,075.47 Contour x1

A Constrained Problem Optimal Solution for Constrained Problem x2 x1
250 200 150 100 50 Constrained Optimum (92.45, 71.32) Profit = $6,075.47 $6,075.47 Contour x1

A 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.

Local 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.

Multiple Local Optima Nonlinear 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.

Multiple 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.

Multiple Local Optima

Single 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.

Single Local Optimum

Single 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.

Single Local Optimum Y X Z 2 4 -2 -4 40 20

Constructing 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.”

Example: Constructing an Index Fund
Lymann Brothers Investments Lymann Brothers has a substantial number of clients who wish to own a mutual fund portfolio that closely matches the performance of the S&P 500 stock index. A manager at Lymann Brothers has selected five mutual funds (shown on the next slide) that will be considered for inclusion in the portfolio. The manager must decide what percentage of the portfolio should be invested in each mutual fund.

Mutual Fund Performance in 4 Selected Years Annual Returns (Planning Scenarios) Mutual Fund Year 1 Year 2 Year 3 Year 4 International Stock Large-Cap Blend Mid-Cap Blend Small-Cap Blend Intermediate Bond S&P

Define the 9 Decision Variables IS = proportion of portfolio invested in international stock LC = proportion of portfolio invested in large-cap blend MC = proportion of portfolio invested in mid-cap blend SC = proportion of portfolio invested in small-cap blend IB = proportion of portfolio invested in intermediate bond R1 = 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)

Define the Objective Function Min (R1 – 13)2 + (R2 – 12)2 + (R3 – 7)2 + (R4 – 2)2 Define the 6 Constraints (including non-negativity) 25.64IS LC MC SC IB = R1 27.62IS LC MC SC IB = R2 5.80IS LC MC SC IB = R3 - 3.13IS LC MC SC IB = R4 IS + LC + MC + SC + IB = 1 IS, LC, MC, SC, IB > 0

Optimal Solution for Lymann Brothers Example R1 = (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

Lymann Brothers Portfolio Return vs. S&P 500 Return Scenario Portfolio Return S&P 500 Return

End of Chapter 12

Slides by . John Loucks St. Edward’s Univ..

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