Excel Tools: Solver and Goal Seek

A Brown Bag discussion for N-8126 Sept 2012

THIS PRESENTATION IS UNCLASSIFIED

Purpose

• This Talk promises to:– (re)introduce some powerful tools in Excel• Optimization – centric functions• Goal seek• Solver

– To show practical examples of how they may be used

• This Talk will not:– Be a course in Linear Optimization

References

• Albright, S. C. (2007) VBA for modelers Thompson – Brookstone.

• Walkenbach, J Microsoft Excel 2010 Bible Wiley.• Ragsdale, C. Decision Analysis with

Spreadsheets• Balakrishnan, N. Managerial Decision Modeling

with Spreadsheets• Frontline Systems: www.solver.com

Block 0: Goal Seek

• Goal Seek: A Line Search algorithm that iteratively seeks to find the (User set) goal value.– Goods: Robust, fast, and easy to use. Can work on non-

linear problems– Bads: Goal Seek is not an optimization algorithm, but

rather a regulated form of iterated guessing. Sometimes doesn’t work. Can miss multiple/optimal solutions

– Others: Will only take literals as the goal value.– For Nerds: Is it Binary or Golden Section? Microsoft

doesn’t specify in documentation

The important thing to know about Goal Seek:

• G/S simply automates the ‘trial – and – error’ approach to problem solving.

• In strict terms, Goal Seek is Guessing.

Invoking Goal Seek

• Data -> What If Analysis -> Goal SeekSet Cell: The ‘target’

The desired value of the Set Cell. Must be

a ‘literal’.

“By Changing”

Goal Seek Example

• Given a maintenance department with two ‘shops’, if aircraft break at a rate of 1 per day, at what rate does each shop need to fix planes in order to have 80% chance that at any given moment there are no down planes?

• This is a queuing problem

Solving

• The hard way: Break out your stochs textbook, and try to formulate this as a Markov Chain

• The Easy way: Q.xls and Goal Seek!

Goal Seek Example

Goal Seek Example II: Military Application

• Recall Lanchester’s Aimed Fire model of combat from our last lunchtime series:

• Given , find the value of that results in 70 blue survivors when the Red side is eliminated.

dBR

dtdB

Bdt

0 0.05; 100; 100B R

Approach 1

TOO EASY! Just solve:

0 0 0 0

0 0 0 0

1 1( )

2 2

1 1( )

2 2

B R B R

B R B R

t tB B

R R

t tR R

B B

B t B R e B R e

R t R B e R B e

Approach II

• Use Goal Seek!

Summary

• Goal seek is ideal in situations where you want answers, not proofs!

• DISCLAIMERS:– You will not know if there are multiple solutions– You will not get any sensitivity analysis

SOME USEFUL EXCEL FUNCTIONS

Named Cells

• Sometimes, you want to use cells, but D$3$ can become cumbersome to type.

• Also, you might want to be able to understand the formulae without an interpreter.

• Naming Cells is very handy for this!• How to? – Use the Name box!

Naming Cells

Name Box

Which is easier to understand?

Formula Auditing

• Sometimes, you want to know what the ‘goezintas’ for a particular answer are. There’s a great function for this: Formula Auditing– Formulas -> Formula Auditing

Example

I cannot overstress how useful this is for reviewing someone else’s spreadsheet!

SUMPRODUCT()

• Sometimes you have data in two columns, and you wish to find the total the product of the two sides.– Such as finding the expectation and variance of a

discrete variable manually– Tallying up costs by aircraft type

• Excel has a special function for this: SUMPRODUCT()

SUMPRODUCT() II

• Takes two arrays, X and Y as inputs – Must be equally sized

• Returns a single number, which is:

• In English: Multiply all these by all those and add up• Very useful for the objective function of a LP.

i ii

X Y

SUMXMY2()

• Sometimes you want to find the squared Euclidian distance between two points or sets of points.– Such as distance between ships in the example we

will work later– Or OLS (Ordinary Least Squares)

• SUMXMY2:– Why did they name it that? I don’t know!– SUM of X Minus Y Squared (2)

SUMXMY2() II

• Takes two arrays, X and Y and returns the sum of square differences (i.e. squared Euclidian measure)

2( )i ii

X Y

Block 1: Solver

• What is it: an optimization package. Can handle LP, MIP, NLP. It’s not anyone’s favorite package, but it’s the one we have.

• Goods: Graphical interface with spreadsheets can make formulation more visually appealing

• Bads: There are some problems where it fails to converge. Limits on number of variables, and number of Integer/Binary variables

• Others: Some important things ‘under the hood’ that should be considered (upcoming slides)

• For Nerds: I’ve never used Evolutionary Solver mode.

Invoking Solver

• To use Solver, it must be installed– Why? Because Bill Gates Hates you, that’s why!

• To use Solver in VBA, it must be installed there as well– Why? See above!

Installing Solver – Main Body

Installing Solver- VBA

Block II: How to use Solver

Solver Differences - 2010

• Excel 2010 has a different (and not better) interface.

• Report options are now available after the solution is complete– Used to be an option in the dialog box

Report Options

A Toy Force Structure Problem

• You need to put together a force that is able to carry 15 helicopters, using a mix of ships at a minimum cost.

• Data Given:

Use at least 1 LHA, 2 LCS, and no more than 4 of any type of ship.

Ship Type Aircraft per Ship Cost Per Steaming DayDDG 1 1000FFG 2 1000CG 2 3000LHA 8 5000LCS 1 1000

Switch to Spreadsheet

Some Pointers

• I like to use color:– Objective function GREEN– Constraints RED– Variables BLUE

• This makes it easy to see ‘what’s what

Sensitivity ReportMicrosoft Excel 14.0 Sensitivity ReportWorksheet: [Book1]Sheet1Report Created: 9/23/2012 7:52:46 PM

Variable CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$K$8 Number Ships 0 500 1000 1E+30 500$K$9 Total A/C Number Ships 2.5 0 1000 250 1000$K$10 Needed Number Ships 0 2000 3000 1E+30 2000$K$11 Number Ships 1 1000 5000 1E+30 1000$K$12 Number Ships 2 500 1000 1E+30 500

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$E$9 Total A/C Cost Per Steaming Day 15 500 15 3 5

Okay, well, that answer was great…(but what does .5 ships mean?)

• In order to go forward, we need to solve as a Mixed Integer Program.

• In Excel, we add another constraint…

… and a whole universe of complexity

WARNING! It is NOT GENERALLY true that the solution to a problem with Integer constraints is ‘close’ to the associated Linear Program!

Solver – Pro Tips

• Frontline solvers claims that the limit on variables is 200, – But it might crash much, much sooner!

• Solutions:– A. Upgrade solver– B. Use MATLAB solver– C. Ask us (NPS) for help.

Solver: what can go wrong

• Here’s something that can go wrong:– Max X^2, unconstrained, with an incumbent

solution of zero.

Practical Example: Optimum placement of Ships

• http://puzzlor.editme.com/Relief-Mission

Problem Description

• Given the map shown and the ability to place two ships, what is the optimum placement to minimize the maximum distance to a target?

• Real-world applications:– Ballistic Missile Defense– Medical Facility Placement– Logistics Hubs– Others?

First, an easier Problem

• What if there was only one ship to be placed?

, ,

2 2

min

. .

( ) (Y-y )

0 10

0 10

X Y Z

i i i

z

s t

Z X x

X

Y

Now, the Placement Problem

• This is actually two problems– If we knew which ship was associated with each

Target , this would be easy– If we knew where the two ships were stationed, it

would be easy to know which Target to associate with them

• We have to figure both out, simultaneously• This is a hard problem – but one that is

probably worth the effort

Formulation

2 21

, , ,

2

1

22 2

( ) ( ) (1 )*

( ) ( ) ( )*

[0,1]

min

. .

i i i

X Y Z S

i i i

X x Y y S BigPen

X x Y y S BigPe

t

Z n

Z

s

S

Z

Why does this work?

• Because for at one and only one of the constraints, the RHS is negative, and therefore irrelevant.– Here we’re using constraints to shape the objective

function!

• While combining the constraints into an expression like looks good, it is highly non-linear and may cause solver to crash

1 2) (1 )*(* )( i ix S xS XX

Summary

• Today we talked about:– Excel Practice, generally– Goal Seek: A useful way for getting answers but

Proofs– Excel Functions:

• Named cells are infinitely better than R$2$:D$2$, etc.• SUMPRODUCT()• SUMXMY2()

– Solver– Practical Applications.

Final Note: Presentation Don’ts

• BEWARE what MS office does when you copy and paste!– If you copy and paste a graph from a spreadsheet, you might

be pasting the entire spreadsheet. This may be handing someone else your entire analysis.

– Solution: Be sure to paste graphs as pictures• BEWARE what MS office does when you use track

changes.– You might leave them in. – Solution: Copy your text, paste to clipboard, then copy and

paste back to word• Solution to both problems: convert to .pdf before

distribution

Point of Contact

• CDR Harrison Schrammhcschram@nps.eduschrammhc@nps.navy.smil.mil831.656.2358

• After April, 2013– BE922