Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 8-1 Nonlinear Programming & Evolutionary Optimization Chapter 8

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 8-2 Introduction to Nonlinear Programming (NLP) u An NLP problem has a nonlinear objective function and/or one or more nonlinear constraints. u NLP problems are formulated and implemented in virtually the same way as linear problems. u The mathematics involved in solving NLPs is quite different than for LPs. u Solver tends to mask this difference but it is important to understand the difficulties that may be encountered when solving NLPs.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 8-3 Possible Optimal Solutions to NLPs (not occurring at corner points) objective function level curve optimal solution Feasible Region linear objective, nonlinear constraints objective function level curve optimal solution Feasible Region nonlinear objective, nonlinear constraints objective function level curve optimal solution Feasible Region nonlinear objective, linear constraints objective function level curves optimal solution Feasible Region nonlinear objective, linear constraints

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 8-4 Local vs. Global Optimal Solutions A C B Local optimal solution Feasible Region D E F G Local and global optimal solution X1X1 X2X2

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 8-5 Comments About NLP Algorithms u It is not always best to move in the direction producing the fastest rate of improvement in the objective. u NLP algorithms can terminate a local optimal solutions. u The starting point influences the local optimal solution obtained.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 8-6 Comments About Starting Points u The null starting point should be avoided. u When possible, it is best to use starting values of approximately the same magnitude as the expected optimal values.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 8-7 Location Problems u Many decision problems involve determining optimal locations for facilities or service centers. For example, –Manufacturing plants –Warehouse –Fire stations –Ambulance centers u These problems usually involve distance measures in the objective and/or constraints.  The straight line (Euclidean) distance between two points ( X 1, Y 1 ) and (X 2, Y 2 ) is:

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 8-8 A Location Problem: Rappaport Communications u Rappaport Communications provides cellular phone service in several mid-western states. u The want to expand to provide inter-city service between four cities in northern Ohio. u A new communications tower must be built to handle these inter-city calls. u The tower will have a 40 mile transmission radius.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 8-9 Graph of the Tower Location Problem Cleveland Akron Youngstown Canton x=5, y=45 x=12, y=21 x=17, y=5 x=52, y= X Y 0 10

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning Defining the Decision Variables X 1 = location of the new tower with respect to the X-axis Y 1 = location of the new tower with respect to the Y-axis

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning Defining the Objective Function u Minimize the total distance from the new tower to the existing towers MIN:

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning Defining the Constraints u Cleveland u Akron u Canton u Youngstown

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning Implementing the Model See file Fig8-10.xls

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning Analyzing the Solution u The optimal location of the “new tower” is in virtually the same location as the existing Akron tower. u Maybe they should just upgrade the Akron tower. u The maximum distance is 39.8 miles to Youngstown. u This is pressing the 40 mile transmission radius. u Where should we locate the new tower if we want the maximum distance to the existing towers to be minimized?

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning Implementing the Model See file Fig8-13.xls

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning Comments on Location Problems u The optimal solution to a location problem may not work: –The land may not be for sale. –The land may not be zoned properly. –The “land” may be a lake. u In such cases, the optimal solution is a good starting point in the search for suitable property. u Constraints may be added to location problems to eliminate infeasible areas from consideration.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning Optimizing Existing Financial Models u It is not necessary to always write out the algebraic formulation of an optimization problem, although doing so ensures a thorough understanding of the problem. u Solver can be used to optimize a host of pre- existing spreadsheet models which are inherently nonlinear.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning A Life Insurance Funding Problem u Thom Pearman owns a whole life policy with surrender value of $6,000 and death benefit of $40,000. u He’d like to cash in his whole life policy and use interest on the surrender value to pay premiums on a a term life policy with a death benefit of $350,000. Year Premium$423 $457 $489 $516 $530 $558 $595 $618 $660 $716 u The premiums on the new policy for the next 10 years are: u Thom’s marginal tax rate is 28%. u What rate of return will be required on his $6,000 investment?

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning Implementing the Model See file Fig8-22.xls

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning The Portfolio Optimization Problem u A financial planner wants to create the least risky portfolio with at least a 12% expected return using the following stocks. Annual Return YearIBCNMCNBS 111.2%8.0%10.9% 210.8%9.2%22.0% 311.6%6.6%37.9% 4-1.6%18.5%-11.8% 5-4.1%7.4%12.9% 68.6%13.0%-7.5% 76.8%22.0%9.3% 811.9%14.0%48.7% 912.0%20.5%-1.9% 108.3%14.0%19.1% 116.0%19.0%-3.4% %9.0%43.0% Avg7.64%13.43%14.93% Covariance Matrix IBCNMCNBS IBC NMC NBS

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning Defining the Decision Variables p 1 = proportion of funds invested in IBC p 2 = proportion of funds invested in NMC p 3 = proportion of funds invested in NBS

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning Defining the Objective Minimize the portfolio variance (risk).

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning Defining the Constraints u Expected return p p p 3 >= 0.12 u Proportions p 1 + p 2 + p 3 = 1 p 1, p 2, p 3 >= 0 p 1, p 2, p 3 <= 1

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning Implementing the Model See file Fig8-26.xls

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning The Efficient Frontier %10.50%11.00%11.50%12.00%12.50%13.00%13.50%14.00%14.50%15.00% Portfolio Return Portfolio Variance Efficient Frontier