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ARIES Portfolio Optimizer User Guide

Copyright 2005 by Landmark Graphics Corporation



TABLE OF CONTENTS

Overview ……………………………………………………………………………. 3Opening Portfolio Optimizer ……………………………………………………..…. 4Components of the Portfolio Builder ……………………………………………..…. 7

Portfolio Building with the Portfolio Optimizer ………………………………...… . 7Setting Up Objective using “Define Objective” Tab ………………………… 8Setting Up “Goals & Constraints” …………………………………………. 10Selecting the Opportunity Set for the Portfolio …………………………….. 12

Mode and Participation Interests ………………………………………… 14Mutually Exclusive Groups ……………………………………………… 15

Setting Up Run Mode and Parameters ……………………………………… 16Portfolio Building with ‘User Defined’ option ……………………………………... 20Function of the Portfolio Manager ………………………………………………….. 22Analyzing Portfolios Using Portfolio Manager’s Reports&Graphs ………………… 23

Reviewing Portfolio Manager – Reports ……………………………………. 24

Cash Flow Elements (Non_Time) Report ……………………………….. 25Cash Flow Elements (Time Series) Report ………………………………. 29Portfolio Comparison Report …………………………………………….. 30Summary of Optimization Problem ……………………………………… 31

Reviewing Portfolio Manager – Graphs ……………………………………. 32Portfolio Performance Graph – Time Series….…….…………………….. 33Portfolio Performance Graph – Non-time Series ………………………….34Portfolio Comparison – Asset Selection Graph …..………………………..34Portfolio Comparison – Cash Flow Elements / Overlay Graph …..………. 36Portfolio Comparison – Cash Flow Elements/ Cross Plots – Non-Time and

Time Series Graphs …..………………………..37

Portfolio Comparison – Goals/Targets vs Portfolio Value – Time Series ....38

Conclusions …………………………………………………………………………...40Appendix – SPE Paper 82009…………………………………………………………41

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OVERVIEW;

ARIES Portfolio Optimizer uses all the information from multi-dimensional data cube build with The ARIES Portfolio - Cube Tools and it helps you ask the right questions tofind the optimal choice of investment opportunities. It gives you ability to run ‘what if “

scenarios and identify tactical decisions in alignment with your goals, strategies andidentified investment opportunities.

It uses state of the art portfolio optimization algorithm (OptQuest) and evaluatesacceptability of projects based on its contribution to the portfolio as a whole andidentifies project mix that provides the corporation with the optimal risk and returnattributes given management’s strategy, risk preferences and capital limitations.

Users could also create and test different scenarios by assigning participation interest tothe projects of choices manually using the option “User Defined” under the part of“Define Objective” tab in the Portfolio Builder.

The Workflow for the ARIES Portfolio is as follows;

Open ARIES Portfolio Module, and enter security login

Goto the Tools, Portfolio Optimizer

Select the cube that you want to work with and connect to the express server by selectingthe express server and entering the user name and password for the server.

Build portfolios either by assigning participation interest to the projects of choicesmanually. (Select” User Defined”) or by using Portfolio Optimizer (Select“Optimization”).

Use manual portfolio building or “User Defined” to test “what if scenarios” and analyzethe volatility of cash flow elements for each scenario created.

Use “Optimization” to determine best project mix and volatility of cash flow elements forthe optimized portfolios. (For the theory of the state of the art optimization algorithm,

see the appendix)

Use Portfolio Manager – Reports/Graphs to analyze optimized portfolio.

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OPENING PORTFOLIO OPTIMIZER;

In the ARIES Portfolio Module, goto Tools, Portfolio Optimizer;

Select the portfolio data cube that you want to work with. You will use this cube for

selecting opportunities to create your portfolio;

On the “Select the Cube you want to work with” dialog box , select the cube byhighlighting it.

On the connection properties dialog box;

- Select the express server that is going to be used- Then, enter the USER ID and password for the server ;

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- Click OK to confirm the selection of the express server.

The ARIES Portfolio Optimizer; Portfolio Builder and Portfolio Manager willthen show up on the screen.

In the ARIES Portfolio Optimizer, there are two main tabs; “Portfolio Builder”, “and“Portfolio Manager”.

The Portfolio Builder is the place where you are going to define your optimization problem. You could define an optimization problem either using “User Defined” bymanually assigning the participation interests to the projectsOr “Optimization” option to use optimizer to define participation interests.

Under Portfolio Builder, there are four main tabs; “Define Objective”, “Set Goal(s)&Constraint(s)”, “Select Data”, and “Run Optimization’’.

When you select to use “Optimization”, first step is to define optimization problem byselecting “Define Objective” and “Set Goal(s) &Constraint(s)” tabs. Next step is toselectThe data for optimization and to set project level specifications at optimization such as“must do, optional, omit” flags, or mutually exclusive groups among the projects using“Asset Selections” tab. You should then define the run mode and parameters for runningoptimization at “Run Optimization “tab as final step.

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When you select to use” User Defined” option to build a manual portfolio, you shoulddefine the variable of interest as objective measure, introduce global goals and targets,and then select the data and assign participation interest in the “Asset Selection” tab.

Even though you are not using optimizer, you should still use “Run Optimization “tab tosave and to give a description to the manual portfolio.

Under the “Portfolio Manager”, there are two main tabs; “Portfolio(s)”, and “Report(s)&Graph(s)”.

The “Portfolio(s)” tab is the place where all your different scenarios/portfolios are saved.When you build different portfolios, you could select to pursue in two ways; either youcould save portfolios and run optimizer later, or you could run optimization when you build the portfolio. If you would like to run them later as batch job, you could use“Portfolio(s)” tab and select the portfolios to be run and click on the “Optimize” tab. If

you would like to open up an existing optimization problem, you could use “Portfolio(s)tab and click on the “Load”. This function will load existing portfolio with all the specs.

When you run optimization, results are saved in the database. You could then use“Report(s) &Graph(s)” part of “Portfolio Manager” and display, compare and contrastdifferent portfolio(s)’s results. All the given report(s)/graph(s) templates could easily bemodified to include different measures and dimensions of portfolio. Their format couldeasily be changed and all components could be analyzed on the fly.

In the next section, we will introduce components and workflows of building andoptimizing, and analyzing portfolio(s).

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COMPONENTS OF THE PORTFOLIO BUILDER

Portfolio Builder is the first tab shows up when you open the ARIES Portfolio Optimizer.It is the place where user could select”User Defined” or “Optimization” option for building portfolios. There are four main components of Portfolio Builder; “Define

Objective”, “Set Goals&Constraints”, “Select Data”, and “Run Optimization”.

To be able to select either “User Defined” or “Optimizer” to define objective function,click on the “Define Objective” Tab, and make your selection by clicking on the radio button;

In the next section, we will introduce components and workflows of portfolio building byusing portfolio optimizer.

PORTFOLIO BUILDING with the PORTFOLIO OPTIMIZER

In this section, you will use optimizer to select best opportunity mix. You will firstdefine your optimization problem by selecting an objective measure. You will then setgoals and constraints at the portfolio level and select your opportunity set for portfolioand define project level requirements. An optimum solution will then be determined byoptimizer given the conditions specified; you could then use reports and graphs toanalyze the optimized portfolio.

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Setting up Objective Using “Define Objective” Tab

In every optimization problem, you should first define whether the optimization problemis a minimization or maximization problem by clicking the choice from the radio buttons.

You should then select the decision variable as objective function measure. In theARIES Portfolio Optimizer, since you have a dynamic connection to the portfolio leveldata with all granularities captured in the ARIES Portfolio cube, you could choose anyelements of your cash flow as decision variable.

To select the measure from the cube data, click on the single arrow key.

Clicking on single arrow key brings “Measure Selector” box with the drop down box ofgroupings of cash flow elements the same as the cube. In this example, let’s select AFITPW at RATE 3# as our decision variable.

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The objective measure is then displayed as follows;

You could also define a ratio as a decision variable if this is your choice by clicking onratio and selecting related numerator and denominator elements.

If your objective is related to the data in certain year, you could also select a decisionvariable with specified year.

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Setting Up “Goals &Constraints”

After setting up objective function, click on “Set Goal(s) & Constraint(s)” tab. At thefollowing display, you could enter the measures (variables) for goals/constraints. Thesevariables could be time series, non-time series, or ratio type of data. Again, you could

choose any elements of your cash flow as goals/constraints since you have dynamic linkto ARIES Portfolio Cube and all its data. To select the measure, click on the single arrowkey.

If you are entering goals/constraints for each year, enter the year in the box given for“Start Year” or select the year from the drop down box. Then select the bound dependingon the nature of the targets /constraints (Min or Max), and enter the value for it in the boxgiven below the bound.

For example, if you have capital constraint, you will have “maximum bound” for a givenamount of available budget. (Example below shows a max of $18MM Capital.) With themaximum bound, optimizer will find a solution for this variable between minus infinityto the value entered as maximum bound if there is solution. If there is a productiontarget, you should at least meet the target so that “minimum bound” will be the choice ofselection. With the minimum bound, optimizer will find a solution for this variable between minimum levels (value entered) to positive infinity if there is solution.

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You could also define a percentage increase or decrease for a selected variable with aninitial value given at the start year over a selected period of time. In the followingexample, a minimum requirement of a 2% increase in appraised sold volume for oil overnine years starting from year 2002 and ending at year 2010 is introduced as follows;

After data entered, click on “ADD” to write goals/constraints to the inputs files of

optimizer. Data is also summarized on the screen as follows;

For changing an entry, in the summary part of the screen (upper pane), highlight the line,this brings the existing entry and then make the change and click on “Modify” to confirmthe change on the screen.

To delete, you should also highlight the line and click on “DELETE” button.

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Selecting the Opportunity Set for the Portfolio

To determine the data set for the portfolio optimization click on the “Select Data” tab.Following display then pops up on your screen.

User is able to use any level of organizational structure with the preferred scenario foroptimization. You might select to include all properties or part of them in the portfolioand in this case then decision level in the portfolio optimization will be property level.You could also select aggregated data for portfolio optimization such as field level, or business unit level.

In the screen above, the left pane shows all cube data. If you do right click on mouse andselect expand list, you could show how the data is laid out in the cube according toorganizational tree. If you would like to collapse the tree, click on “Collapse”.

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You could also select all or partial data for the portfolio optimization by either rightclicking on “Select” or clicking on single arrow key (>). If you would like to remove a property or any level, highlight it then use left single arrow (<) key) to take it out from portfolio optimization or right click and click on “remove”. You could also use leftdouble arrow key to empty the selected properties from optimization and start over again

(<<) or right click and click on “remove all”.

The right pane below shows the selected opportunity set for the portfolio optimization.

In the selected opportunity set, you could also introduce opportunity specific constraints.There are three types of opportunity specific information that you could input such asmode, ranges for participation interest and mutually exclusive groups as summarized thescreen below:

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Mode and Participation Interests:

The mode shows the status of each opportunity in the opportunity set of the optimization.When clicking on the drop down box for mode, there are three options; “optional”, “mustdo”, and “omit”. “Optional” is the default choice for each opportunity in the

optimization with the ranges of min of 25% and maximum of 100% unless otherwisedefined. When optional is selected for the project, the optimizer could either select thatopportunity or not depending on the conditions of optimization problem. If “must do “ isselected for an opportunity, you could either select the default range for participationinterest for the opportunity, or you could even define a certain participation interest to project by setting up min and max level to the same number. Optimizer will then always pick up that opportunity in the solution with given participation interest or with a participation interest in the range selected. Selecting omit will result in leaving out theopportunity from the solution.

Hints: To assign same flags for multiple projects, highlight multiple projects by holding

control key and assign the mode and range then click on update. After all the selectionsmade, click on “View” to see the all selected choices. Following report then shows upsummarizing all the entries.

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Mutually Exclusive Groups;

In the opportunity set, if there are surrogates, or different versions of same opportunitiesrepresenting delays, some technological enhancements, you could create mutuallyexclusive groups for these opportunities. Optimizer is then going to select only one

opportunity from each mutually exclusive group in the solution in addition to otherselected opportunities.

Hint: To create mutually exclusive groups, build surrogates or different scenarios of samemodel and include in your cube. In the right pane, highlight multiple projects of mutuallyexclusive group and select “yes” to confirm creating mutually exclusive group, and givea name for the group in the space given next to group and click on “Update”. When youcreate a group, this group will then be displayed on the drop down box to modify orupdate.

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After creating mutually exclusive groups, click on “view” to see the summary of allopportunity specific information that is going to be used in the optimization.

Setting up Run Mode & Parameters;

After defining optimization problem by selecting objective function, goals/constraints,and data, you should set up the run mode and parameters by clicking on the tab “RunOptimization”.

Run mode defines the range in which optimizer will look for a solution. Lower limitrepresents the range from a given or defaulted value to positive infinity.Upper limit represents the range from minus infinity to given or defaulted value.

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After describing the run mode, you could define the “Run Parameters” to run optimizer.Since a simulation optimization approach (OptQuest®) is used in the optimizationalgorithm, you could run any optimization problem either by choosing time or number ofiterations. Optimizer is then start to a solution with a vector of combinations of theopportunities and it goes through every possible solution for each additional iteration or

time period.(See appendix for the details of third part optimization algorithm OptQuest® andmethodology of Opt Quest®).

When automatic stop is selected, optimizer will stop after the number of iterations wherethe objective value is almost the same.

After setting up the run parameter, you should give a description to the portfolio. If youwould like to run optimizations right the way, then click on “optimize”. This functionwill also save the portfolio automatically. If you would like to run optimization later,click on “save”, then optimization problem will be saved with the given description to berun later.

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When you click on “Optimize”, the “Optimization Performance Graph” shows theobjective function value for each iteration dynamically. In the example below,optimization stops at 5000 iterations. If automatic stop was selected, optimizer would gothrough iterations, and it would automatically stop when it could not find much differentresult for the objective value.

When clicked on “optimize”, above screen shows all entries and output during theoptimization at the “Run Optimization” section.

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After total number of iterations or time is reached in the optimization, following report is produced automatically. Optimization Report gives the participation interest for eachopportunity and also gives the contribution of each opportunity in objective value.

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PORTFOLIO BUILDING with the “USER DEFINED” OPTION

In this section, you will use “User Defined” option to create portfolios by assigning participation interest to the opportunities of choices manually. You could then analyze“what if scenarios” and the volatility of cash flow elements for each scenario created.

In this approach, first, you should define the objective by selecting a measure. In theexample below, P/I Ratio – (PWof AFIT NET/ APPRAISED TOTALINVESTMENTW/O RISK) are used as objective measure.

In the user defined (manual) portfolio, you could still input the goals/constraints. Theseare then used in the Reports&Graphs of the Portfolio Manager to show the valuesachieved in the manual portfolio vs. goals&constraints. In the example above, in additionto using P/I ratio as objective, we would like to see this portfolio’s performance against2% increase for gas production. Therefore, following constraints are entered for eachyear between the years 2002 and 2010.

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You could then click on the “Select Data” Tab and build your manual portfolio byassigning participation interests to the opportunities. If you use single arrow key (>),100% will be assigned to the selected opportunities. If you would like assign partialownership, use (1/x) key and “Set Percentage” and click on “ok”.

You could also click on view to see the assigned participation interests for theopportunities.

Portfolio Manager’s, “Reports&Graphs” are then used to test feasibility of these portfolios.

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Functions of the Portfolio Manager;

Under the Portfolio Manager, there are two main sections; “Portfolio(s)” and“Reports&Graphs”.

In the “Portfolio(s)”, there are list of list portfolios either grouped under “Single” whichis built using optimizer or “Manual” which is built manually. Click on “expand’ to seeall available portfolios. Then, highlight the portfolio of interest. You could load this portfolio or delete or rename it.

If the portfolio optimization is not run yet, you could highlight these portfolios and bringto the right pane, and then you could summit a batch optimization for selectedoptimization problems.

In the “Report(s) & Graph(s)”, standard reports/graphs are provided for selected portfolio or group of portfolios. The reports and graphs assure consistency in presentation and reporting and also make it easier to compare portfolio results and definewhat the best portfolio is for the company given goals and constraints and availableopportunities.

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Analyzing Portfolios Using Portfolio Manager’s - Reports & Graphs;

Portfolio Manager “Reports and Graphs” are the outputs of an optimized portfolio(s) or amanual portfolio. For all the reports and graphs, the source data resides at the cube andeither manually assigned participation interests or optimizer’s solution is applied to

source data to produce reports and graphs.

To have access to the reports and graphs, click on the Portfolio Manager, and then clickon the “Report(s) & Graph(s)”. From the list of optimized portfolios, select theoptimized portfolio that needs to be displayed, and then click on “OK”.

When click on “OK’, the “Reports & Graphs” part of the Portfolio Manager is openedwith the summary information about the selected optimized portfolio from the list;

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Following summary information is then displayed for the portfolio of choice;

Reviewing Portfolio Manager – Reports;

To review Portfolio Manager – Reports, click on the reports.

There are four different types of reports available as follows: Cash Flow Elements (non-time), Cash Flow Elements (Time-Series), Portfolio Comparison, and Summary ofOptimization Problem.

These reports will be analyzed in more detail in the following section;

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Cash Flow Elements – (Non-Time) Report;

This report is designed to pull the data from one-line table of ARIES. In the template below, the rows show the selected elements from the one-line table, the columns showthe selected scenarios. Any selection in the reports can be modified very easily. The

selection for rows are made by double clicking on the account and then double clickingon the “list” on the selector box and selecting measures of interest under the economicsummary listing. The selection for columns are also made by double clicking on“Economic Scenario” and double clicking on “List” on the selector box, selectingscenarios of interest under the economic summary listing.

To change the list of cash flow elements, double click on “Account”;

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On the selector box, click on the “list”, and from the “Selected Accounts”, highlightaccounts that you would like to remove and click on “Remove”, then highlight theaccounts needs to be displayed from the “Available Accounts” list and click on “Add”.And then click on “Ok” to confirm the selection.

In addition to above changes, in the selector box for scenarios, only three scenarios areselected and following format is observed for the report.

The format is very flexible both in the reports and graphs. You could drag and drop anyother dimension either under the rows or under the columns. To drag and drop, singleclick on the dimension and hold it and then drop it either under the row or under thecolumn when you see an arrow.

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When the different product types are included under the rows, the format of the report isas follows;

You could also swap the rows and columns or with any other dimension. Click on the

dimension and hold it and bring on top of the dimension to be swapped. When you seethe sign, drop it.)

The rows and columns are swapped in the following example.

Economic Scenario is swapped with “Portfolios” in the default template.

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Cash Flow Elements – (Time Series) Report;

Cash Flow Element- Time Series Report is designed to pull the time series data fromdetail table of Aries database. You could select any optimized portfolio from the dropdown box, and select the time frame and cash flow elements of interests from the “time”

and “account” lists for a scenario and reserve category of the interest. An exampletemplate is as follows;

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Portfolio Comparison Report

With this report, it is easy to compare participation interests for different portfoliooptimizations as well as selected objective measure, level of optimization, scenario.

Comparison of three different portfolios is displayed in the example below;

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Summary of Optimization Problem;

This report summarizes the optimization problem and some of its results. “Selected”column shows the details of optimization problem such as objective function, entries forgoals and constraints, mutually exclusive groups if there is any, and entries for asset level

requirements such as must do, optional, omit flags.

“Calculated” column on the other hand summarizes computed values for optimized portfolios such as return values, and computed values of all the goals and constraints asdisplayed in the example below;

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Reviewing Portfolio Manager – Graphs

To review Portfolio Manager – Graphs, on the menu click on the graphs.

There are two main groups of graphs available for any portfolio.

Portfolio Performance Graphs includes the templates for non-time and time series data.

Portfolio Comparison Graphs includes following subcategories which will be given inmore detail below;

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Portfolio Performance Graph – Time Series

Values of any cash flow element for the selected time period is displayed in the 3Dformat in the template given below as one of the example of portfolio performancegraphs with the time series data. The format of the graphs is flexible. One could right

click on the white area of the graph and find “Graph Type” and then could select anygraph type desired.

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Portfolio Performance Graph – Non-Time Series

Values of selected one-line table elements for the selected portfolios are displayed in the bar graph format in the template given below as one of the example of portfolio performance graphs with non-time series data. The format of the graphs is flexible. One

could select any graph type from the list to change the graph format. The graph couldshow different categories of reserves by choosing different levels from the drop down box for reserve categories.

Portfolio Comparison – Asset Selection Graph;

Asset Selection graph shows the selected participation interests of the projects for each portfolio. You could then easily compare how projects are selected in different portfolios.

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To set of the format of this graph, double click on “PORTFOLIO” dimension and on theselector box, goto the list, and from the list of optimized portfolios highlight the portfolios that you want to compare and click on “select”. You could also double clickon “Organization “and goto the list and Select “all” or partial projects for the comparison.

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Portfolio Comparison – Cash Flow Elements /Overlay Graph;

Cash Flow Elements/Overlay Graph is designed to compare the trend in the cash flowelements for different optimizations.

If you double click on “ACCT” and goto the selector box and double click on “list”, youshould be able to see all the available elements that you captured at the cube as a choiceof selection for display of account.

In the following example, 6 accounts are selected to be compared for 5 different portfolios over 10 years for 9 economic scenarios. Keep in mind that you could drilldown any of the dimensions by selecting from the drop down box related to thedimension.

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Portfolio Comparison – Cash Flow Elements /

Cross Plots- Non-Time and Time Series Graphs

Cash Flow Elements - Cross Plot graphs are designed to overlay result of cross plots ofdifferent portfolios in one space. The data used could be either time series or non-time

series type of data from the cash flow elements captured.

Non-Time Series data is used in the following example ;( Double click on Portfolio andfrom the list, select the choice of portfolios to be compared, Double click on account toselect two cash flow elements; AFIT PW at 12%discount rate, and WI Total Operating.Cost_ AD Valorem Tax)

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Next example shows the comparison of portfolios for Gross Input Schedule for Oil vs.Gross Input Schedule for Gas at year 2004 for base scenario with total reserves.

Portfolio Performance – Goals/Targets vs. Estimated /

Value Graph

In this graph, you could display your goals/constraints versus the values achieved for theoptimized portfolio in the same space.

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In the following example, for the three optimization problems (portfolios), “AppraisedOil Sold “ vs. “ Target Production Growth” which is a minimum requirement of 2% over8 years are displayed. Production growth expected to be as 2% over nine years starting60,000 MBarrels at 1998 and ending at the rate of 70,298 MBarrels at 2006. (Notice thatscale is different for the portfolio values and the target values.) When the results of

three optimization problems are compared, it is concluded that optimizer finds projectmix for each portfolio over 9 years with much higher production than minimum production requirement. On the other hand, when optimizer is not used, the manual portfolio selected using P/I ranking can not meet the minimum production requirementsfor any of the eight years.

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CONCLUSIONS

ARIES Portfolio Optimizer helps managers to make better and more efficient decisions by aligning goals, strategies and opportunities of the company. There is a dynamic link between the multi-dimensional data cube build with The ARIES Portfolio - Cube Tools

and the ARIES Portfolio Optimizer. Due to this seamless integration, it is easy todefine and pull any data for defining the optimization problem

ARIES Portfolio Optimizer uses state of the art portfolio optimization algorithm (OptQuest) and evaluates acceptability of projects based on its contribution to the portfolio asa whole and identifies optimum project mix given corporation’s strategy, risk preferencesand capital limitations. In the Appendix, a SPE paper is attached as reference for thetheory and methodology of the state of the art optimization algorithm, Opt Quest.

Users could also create and test different scenarios by assigning participation interest tothe projects of choices using the option “user defined” under the section “building portfolio”.

Using ARIES Portfolio Optimizer’s flexibilities and reporting and graphing capabilities,you could ask the right questions to find the optimal choice of investment opportunitiesand run ‘what if “ scenarios and identify tactical decisions in alignment with your goals,strategies and identified investment opportunities.

End result will be better and more efficient decisions for the corporations.

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APPENDIX

SPE 82009

Advanced Optimization Methodology in the Oil and Gas Industry: The Theory of Scatter Search Techniques with Simple Examples

J.April1, F. Glover 1, J. Kelly1, M. Laguna1 M. Erdogan2, B. Mudford2, D. Stegemeier 21 OptTek Systems, Inc.2 Landmark Graphics Corporation

Copyright 2003, Society of Petroleum Engineers Inc.

This paper was prepared for presentation at the SPE Hydrocarbon Economics and Evaluation Symposium held in Dallas, Texas, U.S.A., 5–8 April 2003.

This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s).Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material,as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings aresubject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of thispaper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to anabstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paperwas presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A., fax 01-972-952-9435.

Abst ractA primary purpose of senior management is to make decisions around capital allocation that will improvethe performance of the corporation. Portfolio analysis tools are designed to aid senior management in thedevelopment and analysis of portfolio strategies, by giving them the capability to assess the impact on thecorporation of various investment decisions. To date most of the commercial portfolio optimization

packages have been relatively inflexible and are often not able to answer the key questions asked by seniormanagement. In this paper we will present new techniques that increase the flexibility of optimization toolsand deepen the types of portfolio analysis that can be carried out. A new optimization engine, whichcontains state-of-the-art optimization functionality, allows users to simultaneously address financial returngoals, catastrophic loss avoidance, and performance probability. These innovations enable users toconfidently design effective plans for achieving financial goals, by employing accurate analysis based onreal data. Traditional analysis and prediction methods are based on mean variance analysis, which is knownto be faulty. The new techniques take a more sophisticated and strategic direction. State-of-the-arttechnology integrates optimization and simulation techniques into a global system that guides a series ofevaluations to reveal truly optimal investment scenarios. Specifically, the new optimization techniques usean evolutionary approach called scatter search, reinforced by the adaptive memory strategies of tabu search.The software integrates state-of-the-art metaheuristic procedures, including Tabu Search, Neural Networks,and Scatter Search, into a single composite method. We will give some examples that highlight theimportance and flexibility of the techniques applied to E&P portfolios.

IntroductionPortfolio optimization for capital investment is often too complex to allow for tractable mathematicalformulations. Nonetheless, many analysts force these problems into standard forms that can utilizetraditional optimization technologies such as linear and quadratic programming. Unfortunately, suchformulations omit key aspects of real world settings resulting in flawed solutions based on invalidassumptions. In this paper we focus on a flexible modeling and solution approach that overcomes theselimitations.

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BackgroundThe beneficiaries of the technology discussed here include C level executives responsible for decidingcapital investments and accountable for their performance, finance department analysts charged withdeveloping the capital budget analysis and a project portfolio management plan, and technology managersresponsible for planning and implementing projects. Their needs, which provide compelling reasons to usethe technology, are:

• Technology managers and corporate financial executives are dissatisfied with current way theyaddress risk tolerance.

• They are under continual pressure to improve capital investment performance.

• They need technology that improves the understanding of the analysis and clearly identifies thereasons to make specific investment decisions.

• They are concerned that their competition may be adopting a new and more advanced technology.Capital investment within commercial firms is primarily accomplished with traditional analyses that

include net present value analysis and mean-variance analysis. Although there are many methods beingused to enable capital decisions, there are certain conventions that have become standardized throughimplementation practices. Consequently, many organizations use similar methods to evaluate and selectcapital spending options and monitor their performance.

Many organizations evaluate their capital projects by estimating their "net present value." Net presentvalue (NPV) is calculated by projecting the future cash flows the investment is likely to generate,"discounting" the future cash flows by the cost of capital, and then subtracting the initial investment.

According to conventional wisdom, it makes economic sense to undertake projects if their NPV’s are positive. But this does not guarantee they will be funded. Organizations typically take other factors intoconsideration, which incorporate their ability to fund the initial investment given their debt position, theircurrent operating expenses and cash flow positions, and their strategic considerations including financial performance expectations.

Determining how to allocate investment capital in order to maximize returns is a ubiquitous challengewhere approaches to solutions cover a very wide spectrum. In organizations both public and private, thedecisions of committing limited resources to a variety of uses can either strengthen or deteriorate the veryfinancial foundation of the organization itself. On one end of the spectrum, and at the core of sophisticatedfinancial manuals, capital budgeting procedures many times employ traditional operations research theoriesand techniques to guide and support decisions. On the other end, and anecdotally, many executives admitthat selections come down to mustering intuition, combined with seat-of-the-pants “guestimates”, and

peppered with squeaky wheel assignments.Typically, however, what is common in this arena is building models, which employ pro forma plans

centering around measures of the benefits of the investments- the returns, time horizons over which theinvestments are being made, and estimates of the risks or uncertainty involved. The list of measuresexpands to include such considerations as cash flow, cost of capital, market share, etc.

Evaluations of alternatives are made as well in a variety of ways. From one-at-a-time comparisons ofreturns and risks to more sophisticated portfolio optimization and real option theories, organizations run thegamut in the ways they decide to allocate capital. In the companies using these sophisticated methods,which go beyond single project net present value analysis, many portfolio management methods includemean variance analysis.

In a seminal paper in 1952 in the Journal of Finance, Harry Markowitz laid down the basis for themodern portfolio theory (Markowitz, H., 1952). For his path-breaking work that has revolutionizedinvestment practice, he was awarded the Nobel Prize in 1990. Markowitz focused the investment

profession's attention to mean-variance efficient portfolios. A portfolio is defined as mean-varianceefficient if it has the highest expected return for a given variance, or, equivalently, a portfolio is defined asmean-variance efficient if it has smallest variance for a given expected return.

In Figure 1, the curve is known as the efficient frontier and contains the mean-variance efficient portfolios. The area below and to the right of this mean-variance efficient frontier contains various riskyassets or projects. The mean-variance efficient portfolios are combinations of these risky projects.

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Why are mean-variance portfolios important? Decision makers are risk-averse.They prefer portfolios with high expected returns and low risk. Another importantquestion: How is the risk of a portfolio measured? If portfolio returns are normallydistributed, then its risk can be measured by its variance. However, a substantial body ofempirical evidence suggests that actual portfolio returns are not normally distributed

(McVean, J.R., 2000).

If actual portfolio returns are not normally distributed, then variance is not the appropriate risk measurefor a portfolio. If not variance, what is an appropriate risk measure for a portfolio?

Efficient

Frontier

Expected

Return

Variance of ReturnFig. 1- Efficient Frontier

In practice, mean-variance efficient portfolios have been found to be quite unstable: small changes inthe estimated parameter inputs lead to large changes in the implied portfolio holdings. The practicalimplementation of the mean-variance efficient paradigm requires determination of the efficient frontier.This requires three inputs: expected returns of the projects, expected correlation among these projects, andexpected variance of these projects (individually). Typically, these input parameters are estimated usingeither historical data or forecasts. Researchers have found that estimation errors in these input parametersoverwhelm the theoretical benefits of the mean-variance paradigm.

Now, as cracks in the foundation are becoming too conspicuous to ignore and capital budgeting participants have been dedicated to traditional ideas for so long that they are not able to pull away, even atthe expense of policies that severely hamper their financial growth.

Efforts by more progressive analysts to sound the alert about the crumbling structure underlyingmainstream capital budgeting and investment strategies have not been lacking. Still, the best response has

been to cobble together various ad-hoc measures in an attempt to shore up the framework, or erect amakeshift alternative. Recognition that this response is far from ideal has persuaded many to cling to theold ways, in spite of their apparent defects. The inability to devise a more effective alternative has beendue in large part to limitations in the technology of decision-making and analysis, not only in the area ofinvestments but in other areas of business alike, which has offered no reliable method to conquer thecomplexity of problems attended by uncertainty. As a result, the goal of evaluating investmentseffectively, and to account appropriately for tradeoffs between risk and potential return, has remainedincompletely realized and ripe for innovation.

Over the last several years, alternative technologies (methods) have emerged for optimizing decisionsunder uncertainty. The outcome of this development has begun to penetrate planning and decision-making

43



in many business disciplines, making it possible to study viable solutions to models that are much moreflexible and realistic than those treated in the past. In application to the areas of capital budgeting andinvestment, these alternative technologies are being implemented to create portfolio and asset-allocationstrategies to improve performance. Included in these alternative technologies are agent-based modeling for portfolio optimization, genetic algorithms for portfolio optimization, and real options analysis for capitalspending. All of these technologies seek to improve on the traditional methods by introducing moreflexible, robust, and realistic assumptions and providing more powerful and sophisticated analysis andforecasting tools. Companies marketing these alternative technologies include The Bios Group, Insightful,Merak, United Management Technologies, Glomark, and Portfolio Decisions, Inc.

To date the largest penetration for these technologies have been in academic circles while achievingonly a modicum of success in the marketplace. This indicates that commercial applications of alternativetechnologies are still in the early adoption stages.

Optimization MethodsThe complexities and uncertainties in complex systems are the primary reason that simulation is oftenchosen as a basis for handling the decision problems associated with those systems. Consequently,decision makers must deal with the dilemma that many important types of real world optimization problems can only be treated by the use of simulation models, but once these problems are submitted tosimulation there are no optimization methods that can adequately cope with them.

Advances in the field of metaheuristics—the domain of optimization that augments traditional

mathematics with artificial intelligence and methods based on analogs to physical, biological, orevolutionary processes—have led to the creation of optimization engines that successfully guide a series ofcomplex evaluations with the goal of finding optimal values for the decision variables. One of thoseengines is the search algorithm embedded in the OptQuest optimization system developed by OptTekSystem, Inc.. OptQuest is designed to search for optimal solutions to the following class of optimization problems:

Max or Min F( x)Subject toA x ≤ b (Constraints)gl ≤ G( x) ≤ gu (Requirements)l ≤ x ≤ u (Bounds)

where x can be continuous or discrete.

The objective F( x) may be any mapping from a set of values x to a real value. The set of constraintsmust be linear and the coefficient matrix “A” and the right-hand-side values “b” must be known. Therequirements are simple upper and/or lower bounds imposed on a function that can be linear or non-linear.The values of the bounds “gl” and “gu” must be known constants. All the variables must be bounded andsome may be restricted to be discrete with an arbitrary step size.

A typical example might be to maximize the net present value for a portfolio by judiciously choosing projects subject to budget restriction and a limit on risk. In this case, x represents the specific project participation levels F( x) is the expected net present value. The budget restriction is modeled as Ax ≤ b andthe limit on risk is achieved by a requirement modeled as G( x) ≤ gu where G(x) is percentile value. Eachevaluation, of F(x) and G(x) requires a Monte Carlo simulation of the portfolio. By combining simulationand optimization, a powerful design tool results.

The optimization procedure uses the outputs from the system evaluator, which measures the merit of the

inputs that were fed into the model. On the basis of both current and past evaluations, the optimization procedure decides upon a new set of input values (see Figure 2).

The optimization procedure is designed to carry out a special “non-monotonic search,” where thesuccessively generated inputs produce varying evaluations, not all of them improving, but which over time provide a highly efficient trajectory to the best solutions. The process continues until an appropriatetermination criterion is satisfied (usually based on the user’s preference for the amount of time to bedevoted to the search).

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Optimization

Procedure

Input

Output

System

Evaluator

Fig. 2- Coordination Between Optimization and Evaluation

Project Portfolio Optimization In many industries, strategic planning requires executives to select a portfolio of projects for funding thatwill likely advance the corporate goals. In general, there are many more projects than funding can supportso the selection process must intelligently choose a subset of projects that meet the companies profit goalswhile obeying budgetary restrictions. Additionally, executives wish to manage the overall risk of a portfolio of projects and ensure that cash flow and other such “accounting” type constraints are satisfied.

The Petroleum and Energy (P&E) industry uses project portfolio optimization to manage itsinvestments in the exploration and production of oil and gas. Each project’s proforma is modeled as asimulation capturing the uncertainties of production and sales.

The application illustrated here involves five potential projects with ten year models that incorporate

multiple types of uncertainty in drilling, production, and market conditions. We examined multiple cases todemonstrate the flexibility of the software to enable a variety of decision alternatives. We present astandard case and one that utilizes the power of simulation optimization.

Case 1

In case 1, the decision was to determine participation levels [0,1] in each of the five projects with the objective of maximizing expected net present value of the portfoliowhile keeping the standard deviation of the net present value of the investment below aspecified threshold. This is the traditional Markowitz approach. In this case, all projectsmust begin in the first year.

Maximize E (NPV)

While keeping σ ≤ 10,000 M$All projects must start in year 1

In this case, the best investment decision resulted in an expected net present value ofapproximately $37,400 M with a standard deviation of $9,500 M. Figure 3 shows thecorresponding non-normal NPV distribution.

Frequency Chart

M$

Mean = $37,393.13.000

.007

.014

.021

.028

0

7

14

21

28

$15,382.13 $27,100.03 $38,817.92 $50,535.82 $62,253.71

1,000 Trials 16 Outliers

Forecast: NPV

Fig. 3- Case 1 NPV Distribution

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Case 2

The goal was to determine participation levels in each project where starting times for each projectcould vary and we would maximize the probability of exceeding the expected net present value of $47,500M (which was achieved in a previous analysis). Risk was controlled by limiting the 10th Percentile of NPV.

Maximize Probability( E (NPV) ≥ 47,455 M$)

While keeping 10th Percentile of NPV ≥ 36,096 M$All projects may start in year 1, year 2, or year 3

Frequency Chart

M$

Mean = $83,971.65.000

.008

.016

.024

.032

0

8

16

24

32

$43,258.81 $65,476.45 $87,694.09 $109,911.73 $132,129.38

1,000 Trials 13 Outliers

Forecast: NPV

Fig. 4- Case 2 NPV Distribution

In this case, where starting times could vary, and we wanted to maximize the chance of exceedingthe net present value of $47,500 M, the best investment decision resulted in an expected net present valueof approximately $84,000 M with a standard deviation of $18,500 M. The NPV had a 99% probability ofexceeding $47,500 M (see Figure 4). This case demonstrates that adopting measures of risk other than

standard deviation can result in superior portfolios. Simulation optimization is the only technology that canoffer these types of analyses.

Scatter SearchThe optimization technology used here is the metaheuristic known as scatter search. Scatter search has

some interesting commonalties with genetic algorithms (GA), although it also has a number of quitedistinct features. Several of these features have come to be incorporated into GA approaches after anintervening period of approximately a decade, while others remain largely unexplored in the GA context.

Scatter search is designed to operate on a set of points, called reference points, which constitute goodsolutions obtained from previous solution efforts. Notably, the basis for defining “good” includes specialcriteria such as diversity that purposefully go beyond the objective function value. The approachsystematically generates combinations of the reference points to create new points, each of which ismapped into an associated feasible point. The combinations are generalized forms of linear combinations,accompanied by processes to adaptively enforce constraint-feasibility and encourage requirement-feasibility.

The scatter search process is organized to (1) capture information not contained separately in theoriginal points, (2) take advantage of auxiliary heuristic solution methods (to evaluate the combinations produced and to actively generatenew points), and (3) make dedicated use of strategy instead of randomization to carry out component steps.

Figure 5 sketches the scatter search approach in its original form. Extensions can be created to takeadvantage of memory-based designs typical of tabu search (Glover and Laguna, 1997). Two particularfeatures of the scatter search proposal deserve mention. The use of clustering strategies has been suggested

46



for selecting subsets of points in step 2, which allows different blends of intensification and diversification by generating new points “within clusters” and “across clusters.” Also, the solutions generated by thecombination method in step 2 are often subjected to an improvement method, which typically consists of alocal search procedure. The improvement method is capable of starting from a feasible or an infeasiblesolution created by the combination method.

1. Apply a diversification generationmethod to build a starting set ofsolutions. Designate a subset ofthe best points (judged by quality

and diversity) to be reference points.

while (stopping criteria are notsatisfied) {

2. Form combinations of subsetsof the current reference pointsto create new points. Thecombinations are (a) chosento produce points both insideand outside the convex regionspanned by the reference

points, and (b) modified bygeneralized mapping processes to yield feasible points according to theconstraints in the problem(both linear and integralityconstraints).

3. Update the reference set byselecting points that canimprove the quality and/ordiversity of the set.

Fig. 5- Scatter search outli ne

It is interesting to observe similarities and contrasts between scatter search and the original GA proposals. Both are instances of what are sometimes called “population based” or “evolutionary”approaches. Both incorporate the idea that a key aspect of producing new elements is to generate someform of combination of existing elements. However, GA approaches are predicated on the idea of choosing parents randomly to produce offspring, and further on introducing randomization to determine whichcomponents of the parents should be combined. By contrast, the scatter search approach does not

47



emphasize randomization, particularly in the sense of being indifferent to choices among alternatives.Instead, the approach is designed to incorporate strategic responses, both deterministic and probabilistic,that take account of evaluations and history. Scatter search focuses on generating relevant outcomeswithout losing the ability to produce diverse solutions, due to the way the generation process isimplemented. For example, the approach includes the generation of new points that are not convexcombinations of the original points. The new points constitute forms of extrapolations, endowing themwith the ability to contain information that is not contained in the original reference points.

Scatter search is an information-driven approach, exploiting knowledge derived from the search space,high-quality solutions found within the space, and trajectories through the space over time. Theincorporation of such designs is responsible for enabling the optimizer (which is called OptQuest) toefficiently search the solution space of optimization problems in complex systems.

To learn more about scatter search refer to the tutorial articles by Glover (1998), Glover, Laguna andMartí (1999 and 2000), Laguna (2000) and Laguna and Armentano (2001). Recent applications of scattersearch include the linear ordering problem (Campos, et al. 1999 and Campos, Laguna and Martí 1999), permutation problems (Campos, Laguna and Martí 2001), transportation (Corberán, et al. 2000), nonlinearoptimization (Laguna and Martí 2000) and machine scheduling (Laguna, et al. 2000).

The OptimizerThe optimizer seeks to find an optimal solution to a problem defined on a vector x of bounded variables.That is, the user can define several types of optimization problems depending on the combination of

variables:

• Pure continuous

• Pure discrete (including pure binary problems)

• Mixed problems (continuous-discrete, continuous-permutation, discrete- permutation or continuous-discrete-permutation)

Also, the optimization problem may be unconstrained, include linear constraints and/or requirements.The optimizer detects small pure discrete or permutation problems to trigger a complete enumerationroutine that guarantees optimality of the best solution found.

The optimizer employs special mechanisms to search for optimal solutions to problems defined withcontinuous and discrete variables. Similar mechanisms are used to tackle pure or mixed permutation problems and details can be found in Campos, Laguna and Martí (2001). The scatter search methodimplemented in the optimizer begins by generating a starting set of diverse points. This is accomplished bydividing the range of each variable into 4 sub-ranges of equal size. Then, a solution is constructed in twosteps. First, a sub-range is randomly selected. The probability of selecting a sub-range is inversely proportional to its frequency count (which keeps track of the number of times the sub-range has beenselected). Second, a value is randomly chosen from the selected sub-range. The starting set of points alsoincludes the following solutions:

• All variables are set to the lower bound

• All variables are set to the upper bound

• All variables are set to the midpoint x = l + (u-l)/2

• Other solutions suggested by the user

A subset of diverse points is chosen as members of the reference set. A set of points is considereddiverse if its elements are “significantly” different from one another. The optimizer uses a Euclideandistance measure to determine how “close” a potential new point is from the points already in the referenceset, in order to decide whether the point is included or discarded.

When the optimization model includes discrete variables, a rounding procedure is used to mapfractional values to discrete values. When the model includes linear constraints newly created points aresubjected to a feasibility test before they are sent to the evaluator (i.e., before the objective function valueF( x) and the requirements G( x) are evaluated).

48



Note that the evaluation of the objective function may entail the execution of a simulation, and

therefore it is important to be sure to evaluate only those solutions that are feasible with respect to the set of

constraints. For ease of notation, we represent the set of constraints as A x ≤ b, although equalityconstraints are also allowed. The feasibility test consists of checking (one by one) whether the linearconstraints are satisfied. If the solution is infeasible with respect to one or more constraints, the optimizer

formulates and solves a linear programming (LP) problem. The LP (or mixed-integer program, when x contains discrete variables) has the goal of finding a feasible solution x* that minimizes a deviation between

x and x*. Conceptually, the problem can be formulated as:

Minimize+−

+ d d

subject to A x* ≤ b

0*=+−−

+− d d x x

l ≤ x* ≤ u

, ≥ 0−d +d

where and are, respectively, negative and positive deviations of x−d +d * from the

infeasible point x. The implementation of this mechanism within the optimizerincludes ascaling procedure to account for the relative magnitude of the variables and adds a termto the objective function to penalize maximum deviation. Also, the optimizer treats pure binary problems differently, penalizing deviations without adding deviation variables orconstraints. When the optimization problem does not include constraints, infeasible points are made feasible by simply adjusting variable values to their closest bound androunding when appropriately. That is, if x > u then x

* = u and if x < l then x* = l.

Once the reference set has been created, a combination method is applied to initiatethe search for optimal solutions. The method consists of finding linear combinations ofreference solutions. The combinations are based on the following three types, whichassume that the reference solutions are x′ and x ′′ :

x = ′ x − d x = ′ x + d x = ′′ x − d

Where d = r ′′ x − ′ x

2 and r is a random number in the range (0, 1). Because a different

value of r is used for each element in x, the combination method can be viewed as asampling procedure in a rectangle instead of a line in a two dimensional space (seeUgray, et al. 2001). The number of solutions created from the linear combination of tworeference solutions depends on the quality of the solutions being combined. Specifically,

when the best two reference solutions are combined, they generate up to 5 new solutions,while when the worst two solutions are combined they generate only one.In the process of searching for a global optimum, the combination method may not beable to generate solutions of enough quality to become members of the reference set. Ifthe reference set does not change and all the combinations of solutions have beenexplored, a diversification step is triggered (see step 4 in Figure 4). This step consists ofrebuilding the reference set to create a balance between solution quality and diversity. To preserve quality, a small set of the best (elite) solutions in the current reference set is used

49



to seed the new reference set. The remaining solutions are eliminated from the referenceset. Then, the diversification generation method is used to repopulate the reference setwith solutions that are diverse with respect to the elite set. This reference set is used asthe starting point for a new round of combinations.

Conclusions

The approach discussed here brings intelligence to software for corporate decision-making, and gives a newdimension to optimization and simulation models in business and industry. The system empowers decisionmakers to look beyond conventional decision-making approaches and actually pinpoint the most effectivechoices in uncertain situations. The concepts discussed in this paper should allow senior management tomaximize financial return while accurately measuring and controlling risk.

References1. Barr, R. S. 2000a. “Neural Network Design and Training by Adaptive Polynomial Column Generation: A New (LP)-

Optimization Approach,” Technical Report rsb51, Southern Methodist University.2. Barr, R. S. 2000b. “Non-linear Discrimination by Adaptive Polynomial Column Generation,” Technical Report

rsb52, Southern Methodist University.3. Glover, F. 1990. “Improved Linear Programming Models for Discriminant Analysis,” Decision Sciences, Vol. 21,

pp. 771-785.4. Glover, F. 1996. "Tabu Search and Adaptive Memory Programming: Advances, Applications and Challenges,"

Interfaces in Computer Science and Operations Research, R. Barr, R. Helgason and J. Kennington (eds.) KluwerAcademic Publishers, pp. 1-75.

5. Glover, F., J. Kelly, and M. Laguna. 1999. “New Advances Wedding Simulation and Optimization,” in theProceedings of WSC’99, David Kelton, Ed.

6. Glover, F. and M. Laguna. 1997. Tabu Search, Kluwer Academic Publishers.

7. Glover, F. and M. Laguna. 1997a. “General Purpose Heuristics for Integer Programming ⎯ Part I,” Journal of

Heuristics, vol. 2, no. 4, pp. 343-358.

8. Glover, F. and M. Laguna. 1997b. “General Purpose Heuristics for Integer Programming ⎯ Part II,” Journal of

Heuristics, vol. 3, no. 2, pp. 161-179.9. Glover, F., M. Laguna, and R. Marti. 2000. “Fundamentals of scatter search and path relinking,” Control and

Cybernetics, Vol. 29, No. 3, 653-684.10. Kelly, J., B. Rangaswamy, and J. Xu. 1996. “A Scatter-Search-Based Learning Algorithm for Neural Network

Training,” Journal of Heuristics, vol. 2, pp. 129-146.11. Markowitz, H. 1952. “Portfolio Selection,” The Journal of Finance, vol. VII, no. 1, pp. 77-91.

12. McMillan, C., M. C. Mozer, and P. Smolensky. 1992. “Rule Induction Through a Combination of Symbolic andSubsymbolic Processing.” in Advances in Neural Information Processing Systems 4, S. Hanson, L. Lippman, andL. Giles, (Eds.) Morgan Kaufmann.

13. McVean, J. R. 2000. “The Significance of Risk Definition on Portfolio Selection,” paper presented at the 2000 SPEAnnual Technical Conference and Exhibition, Dallas, TX.

14. Roy, A. 2001. “A New Learning Theory and Polynomial-time Autonomous Learning Algorithms for GeneratingRadial Basis Function (RBF) Networks,” in Radial Basis Function Neural Networks 1: Recent Developments in

Theory and Applications, Howlett, R. J. and Jain, L. C. (Eds.) Physica Verlag, Chapter 10, pp. 253-280.15. Roy, A., S. Govil, and R. Miranda. 1995. “An Algorithm to Generate Radial Basis Function (RBF)-like Nets for

Classification Problems,” Neural Networks, Vol. 8, No. 2, pp.179-202.

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