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Introduction to Optimization for Business Problem

Solving

What is Optimization? In a business problem

context

Realism

- Loosely Finding the best solution to a

problem - More precise Finding the answer to a problem

that minimizes (maximizes) some objective or goal

of a decision maker while taking into account

business constraints - Mathematical version Finding the values of a

set of decision variables that minimizes

(maximizes) some objective function subject to

constraints (equations or inequalities) on the

decision variables

An Observation

- Many useful, important problems in business can

be formulated as - Max or Min f(x1,x2 , ...xn) (objective

function) - Subject to f1(x1,x2 , ...xn) ? b1 (1st

constraint) - f2(x1,x2 , ...xn) ? b2 (2nd constraint)
- fm(x1,x2 , ...xn) ? bm (mth constraint)
- xi ? 0 , i1..n, (decision

variables)

f(x) just means some function of x

Some of the toughest mathematical problems solved

routinely in business today are optimization

problems

So, what is it that makes these problems

difficult? Recall the Traveling Salesman Problem

from day one of class.

Starting with a Modeling Challenge

- Modeling, solving, analysis, embedding
- Modeling can be very challenging and can affect

our ability to solve the problem - Download IRSSuperComputers-Shell.xls
- IRS has determined monthly needs for

supercomputing resources for next 12 months. They

can rent computers for either 1, 2, or 3 months.

The per computer costs decline if we rent for

more months. Their problem is to figure out a

supercomputer rental plan that meets their needs

for the next 12 months and that does so at

minimum cost. - They also need to know how their total rental

costs increase as the monthly need for

supercomputers increases.

Optimization problems ? some very clever, very

important, very cool computational methods

- Calculus
- Classic simplex method for solving LPs
- Interior point methods for LPs
- Branch bound, cutting planes for integer

problems - Various heuristic approaches for solving very

difficult combinatorial problems - Simulated annealing, tabu search, genetic

algorithms, ant colony optimization, swarm

optimization

Example 1 Simple 1 week, days-off problem

- Formulated model in Excel and we will solve it

using Solver - Goal 1 give flavor of optimization applied to

scheduling - Goal 2 illustrate fact that scheduling policies

affect staffing needs - Goal 3 real scheduling problems can lead to huge

optimization problems

SchedulingDSS_Northpark.xls

Example 2 Simple 1 day, shift scheduling problems

- Formulated model in Excel and we will solve it

using Solver - Goal 1 see difference between shift and days-off

scheduling - Goal 2 treat staffing requirements as both hard

and soft constraints - Goal 3 real scheduling problems can lead to huge

optimization problems

ShiftSchedulingModel1.xls ShiftSchedulingModel2.x

ls

Just a few examples (of many, many applications)

Data mining

Production Planning

Staff Scheduling

Logistics

Cash Flow Planning

Project Mgt.

Market Research Planning

The Challenge of Staff Scheduling

1

So , how much staff is needed and how should they

by scheduled?

2

3

Recent Examples from Interfaces

- Implementing Large-Scale Optimization Models in

Excel Using VBA Interfaces, July-August 2007 37

370 - 382. - General Motors Optimizes its Scheduling of

Cold-Weather Tests, Interfaces, 34, 5, Sept-Oct

2004 - Improving Volunteer Scheduling for the Edmonton

Folk Festival, Interfaces, 34, 5, Sept-Oct 2004

(Uses Excel Solver and VBA) - GE Plastics Optimizes the Two-Echelon Global

Fulfillment Network at Its High Performance

Polymers Division Interfaces, 34, 5, Sept-Oct

2004 (Uses Excel Solver and VBA) - Optimizing On-Demand Aircraft Schedules for

Fractional Aircraft Operators Interfaces, 33, 5,

Sept-Oct 2005 - UPS Optimizes its Air Network, Interfaces, 34, 1,

Jan-Feb 2004 - NBCs Optimization Systems Increase Revenues and

Productivity, Interfaces, 32, 1, Jan-Feb 2002 - Heery International's Spreadsheet Optimization

Model for Assigning Managers to Construction

Projects, Interfaces Volume 30, Number 6, Nov/Dec

2000 - Design and Use of the Microsoft Excel Solver,

Interfaces Volume 28, Number 5, Sep/Oct 1998 - Lets search for optimization and spreadsheet

at Interfaces site

The OpenCourseWare Experiment at MIT

- http//ocw.mit.edu/index.html
- A few years ago, MIT announced they would make

all of their course materials available to the

public via the web, free of charge - Mass. Institute of Technology just release their

first courseware sites to the public - One of the first pilot courses in the Sloan

School of Management is - 15.053-Introduction to Optimization
- syllabus has nice set of themes about

optimization in business - the entire course notes and materials are

publicly available - found a nice link from there to a web site,

LP-Explorer, that shows graphical solution of

2-variable linear programs using the simplex

method - Other courses
- 15.073J-Logistical and Transportation Planning

Methods (urban operations research) includes

500 page online textbook - 15.094-Systems Optimization Models and

Computation

Making LPs Practical for Business

- Mathematical theory
- WWII
- Computers
- George Dantzig discovers Simplex Method
- Tons O research
- LINDO, CPLEX, GAMS, AMPL, GNU GLPK
- Terrific applications in industry
- Whats Best? optimization spreadsheets
- Solver optimization for the masses
- Example Tactical Scheduling Analysis

Some Optimization Concepts

- A potential solution is feasible if it satisfies

all the constraints we build in the model - a model is infeasible if no solution satisfies

all the constraints (p96) - A potential solution is optimal if it is feasible

AND it is better than all other feasible

solutions in minimizing (or maximizing) our

objective - a model is unbounded (p97) if we can make the

objective as big as we want (assume were

maximizing) and still satisfy the constraints - So, how do we search among the (potentially huge

number of) feasible solutions to find the optimal

solution? - thats what optimization algorithms such as those

built into the Excel Solver do

Big Steps in Doing Optimization Modeling of

Business Problems

DessertPlanning-MI.xls

- Decision variables?
- Objective?
- Constraints?

Formulate Model

- Pick algorithm
- Feasible?
- Optimal, good unbounded?

Solve Problem

- How does optimal solution change for changes in
- problem data
- additional constraints
- relaxed constraints

Sensitivity Analysis

1. Develop or formulate the model

- This is the art and craft of modeling
- Capture essence of the problem but keep the thing

of reasonable size and complexity - Often not clear what are easy and what are

hard - Identify and represent objective function
- Identify represent constraints
- Representing optimization problems
- algebra with paper pencil
- Example p77 of Practical Management Science
- what makes this a linear problem?
- algebraic modeling languages (e.g. AMPL, GAMS,

LINGO) - Example multimip1
- Excel or other spreadsheets
- Solver uses the familiar notions of ranges and

formulas along with custom dialog boxes to

represent optimization problems - Whats Best? (originally developed for Lotus 123)

is another spreadsheet add-in and is available

from Lindo Systems, Inc.

2. Solve the optimization problem

- Depending on the specific nature of the problem,

different algorithms may be used - One variable - A manual line search or hill

climbing - Linear simplex method (Ch 3), interior point

methods - Non-linear (Ch 7) calculus based methods
- Integer, linear (Ch 6) branch and bound
- Non-linear, integer (Ch 8) branch and bound

heuristics such as genetic algorithms, tabu

search, simulated annealing, ant colony, swarm - Solver does a nice job of choosing an appropriate

algorithm - May find our formulation has no feasible solution
- Our constraints cannot all be satisfied
- We have made a modeling error that must be fixed
- May or may not be able to find a provably optimal

solution - Depends on the specific mathematical problem we

are solving and on the solution algorithm used - Often well be satisfied with a good solution
- Must remember that model likely based on numerous

simplifying assumptions

About optimization in PMS Well visit these

during the term

- Linear programming Ch 3 and Ch 4
- objective and constraints must be linear
- Integer programming Ch 6
- when you want to force some variables to take

integer values only - useful for yes/no type problems (investment

planning, facility location, routing, scheduling) - Non-linear models Ch 7
- many real problems like pricing contain

non-linear elements - many marketing models and finance models
- Genetic or evolutionary algorithms Ch 8
- just a different way of solving many kinds of

optimization problems - can handle all kinds of bizarre formulas

containing Excel functions like Min(), Max(),

If(), Abs() - based on analogy with evolution of life (genes,

mutations, offspring, etc.)

Some Mathematical Programming Solver Tutorials

and Optimization Resources

- e-Optimization.com
- A High-Level Look at Optimization Past, Present

and Future - http//www.frontsys.com/tutorial.htm (Frontline)
- Practical Optimization A Gentle Guide
- http//www.sce.carleton.ca/faculty/chinneck/po.htm

l

Weve already done some optimization

- Break even
- decision variable?
- what was our objective in using Goal Seek?
- constraints?

- Up to this point
- how have we optimized?
- how did we get the value of the objective

function for different values of the decision

variables? - how many decision variables in each problem?
- have we modeled constraints?

- Golf club pricing
- decision variable?
- objective function?
- min or max?
- constraints?

Lets revisit Example 2.2 - Links A typical

pricing problem

Lets revisit Example 2.2 - Links A typical

pricing problem

- Based on ideas in economics and marketing
- demand is influenced by price
- Businesses attempt to set prices to maximize

profit in general - Well explore the basic components of an

optimization problem - Well explore the basic ideas behind trying to

find optimal solutions - hill climbing
- using Solver
- using Data Tables to verify optimal solution
- well add one simple constraint a cliff

Links Pricing Model

- If Links Company charges P dollars per unit, then

its profit will be (P 250)D, where D is the

number of units demanded. - 250 is the per unit cost
- The problem, however, is that Demand depends on

Price. - As P ? D ? and as P ? D ?
- Therefore the first step is to find how D varies

with P the demand function. - In fact, this is the first step in almost any

pricing problem. - Recall we tried linear, exponential, and power

functions and compared them using Mean Absolute

Percent Error.

Power function model

What are a and b and where did we get them? How

do a and b affect the shape of the function?

Pricing Model -- continued

- Lets start with the best fit Power Function
- A constant elasticity demand function of the form

D aPb. - You might recall from economics that the

elasticity of demand is the percentage change in

demand caused by a 1 increase in price. - The larger the (magnitude of) elasticity is, the

more demand reacts to price. The advantage of the

constant elasticity demand function is that the

elasticity remains constant over all points on

the demand curve.

Lets look at Profit function

DDemand, PPrice, CUnit Cost

Notice that Profit is a non-linear function of

price. Why?

Recall that a and b are parameters of the power

function

Heres the pricing optimization problem

- Max aPb(P-Cost) (objective function)
- Subject to
- P ? 0

Price (P) is our only decision variable

Our only constraint is that price must be

non-negative

Lets look at Solver for solving this problem

using a number of techniques. Open

GolfClubsOptimization-InClass.xls.

Solver Main Dialog

Objective function cell a formula

Maximize, minimize or try to hit a target

Attempt to solve

Decision variables

See Options slide

Premium Solver

Build constraints using

Clear dialog box

If you use range names, theyll show up here and

will make model more readable.

About Constraints and Ranges

- Can define constraints using ranges

B16E16ltB18E18 Equivalent to four separate

constraints B16ltB18, C16ltC18, D16ltD18, E16ltE18

- If range name ProducedB16E16 and

CapacityB18E18

ProducedltCapacity Equivalent to four separate

constraints B16ltB18, C16ltC18, D16ltD18, E16ltE18

Solver Options

Solution search time limits

Save model to, or Load model from range on

spreadsheet

Tell Solver problem is linear

Assume all decision variables gt0

Advanced options for non-linear problems

Modeler Beware

- Dangerous to blindly use Solver without

understanding a little about what it does and how

it does it. - For some problems, Solver can guarantee a

globally optimal solution. - concave or convex objective function subject to

linear constraints (see Section 7.2) - For other, very realistic, problems an optimal

solution cannot be guaranteed - hilly objective functions with many peaks and

valleys http//www.projectcomputing.com/resource

s/psovis/index.html - integer constraints, non linear constraints such

as use of IF(), MAX(), MIN(), ABS() functions - Open NastyFunction.xls

Lets throw Solver at this function

Linear Programming

DessertPlanning-MI.xls ProductMix-InClass.xls

LPlinear program

- Many useful, important problems can be formulated

as - Maximize c1x1 c2x2
cnxn (objective

function) - Subject to a11x1 a12x2
a1nxn ? b1 (1st

constraint) - a21x1 a22x2
a2nxn ? b2 (2nd

constraint) - am1x1 am2x2
amnxn ? bm (mth

constraint) - xi ? 0 , i1..n, (decision

variables) - The ci and aij are just numeric coefficients

that are multiplied by the values of the decision

variables (xi)

LP

The Major Assumptions for Linear Programs (Sec

3.5)

- Objective function and constraints are linear

functions of the decision variables - WHAT DOES THAT MEAN?
- is 5x3y linear? (x and y are the variables)
- is 5x23y linear? what about exx? 105xy?
- Decision variables are divisible
- CAN BE FRACTIONAL VALUES
- Practical implication?
- Coefficients in objective function and

constraints are given numbers - NO RANDOMNESS
- Practical implication?

About Linear Programming

- The constraints define the feasible region
- Just a set of linear inequalities
- The shape of the feasible region is a polytope

(in 2D its a polygon and in 3D a polyhedron) - Turns out that the optimal solution will always

lie at a corner of the feasible region - Simplex methods provides an efficient way of

running from corner to corner in the polytope

looking for the best solution - Impact of adding constraints on feasible region

and optimal solution?

We slide the isoprofit line up in a perpendicular

direction until moving it further will result in

it leaving the feasible region. Wherever it

intersects the feasible region last is the

optimal solution (or solutions).

See Section 3.3, p76-78 in PMS for more on

graphical solutions to linear programs

Next Steps

- Lets solve another optimization problem together
- Adverstising Optimization (AdvOpt-Shell.xls)
- More about linear (including integer)

optimization problems - Many business problems require some or all of the

decision variables to be integers. Examples??? - Basic ideas about how linear and integer programs

are solved - read Sec 3.6 (p76-78) if havent already
- read Sec 6.1-6.3 (285-295)
- Well continue Excel based application

development using an optimization modeling example

Yet Another Observation

- Many useful, important problems can be formulated

as - Maximize c1x1 c2x2
cnxn (objective

function) - Subject to a11x1 a12x2
a1nxn ? b1 (1st

constraint) - a21x1 a22x2
a2nxn ? b2 (2nd

constraint) - am1x1 am2x2
amnxn ? bm (mth

constraint) - xi ? 0 , i1..n, (decision

variables) - Some of the xi must be integers

MIP

MIPmixed integer-linear program

About Integer Programming

- Some or all variables restricted to be integers

(or binary) - Can be very difficult to solve given explosion in

problem size - n binary variables ? 2n possible solutions
- Solver uses Branch Bound technique
- a way to implicitly enumerate all the possible

solutions without actually doing it - Subproblems are just linear programs (i.e. the

integer constraints are relaxed) - http//mat.gsia.cmu.edu/orclass/integer/node13.htm

l - If you want to see a nice example of how branch

and bound works

Some Advanced Solver Options

- Max Time Solver will run for X seconds before

pausing - Iterations Solver will make this many

iterations in the solution algorithm details

depend on which solution algorithm is used.

Our Deterministic Optimization World

Linear Models

Non-linear Models

Linear programs easy

Non-linear programs some easy, some really,

really hard

Mixed integer linear programs some easy, some

really, really hard

Non-linear integer programs generally really

tough

Excel Solver for our Deterministic Optimization

World

Non-linear Models

Linear Models

Non-linear programs Standard solver- GRG Premium

solver Genetic alg.

Linear programs Standard solver AlgSimplex method

Non-linear integer programs Branch Bound

Standard solver- GRG or Premium solver

Genetic alg.

Mixed integer linear programs Standard

solver AlgBranch Bound

Some Advanced Solver Options

- Precision Specifies how close a number needs to

be to an integer value to be considered an

integer and how close to constraint values

solutions need to be considered feasible - Tolerance For integer problems, Solver will

stop if best solution so far is within this

specified percentage of known optimal value

The GRG2 (nonlinear) algorithm uses the

Convergence edit box and Estimates, Derivatives,

and Search option button groups.