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## Introduction to Optimization for Business Problem Solving

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Title: Introduction to Optimization for Business Problem Solving

1
Introduction to Optimization for Business Problem
Solving
2
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
• 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

3
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
4
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.
5
Starting with a Modeling Challenge
• Modeling, solving, analysis, embedding
• Modeling can be very challenging and can affect
our ability to solve the problem
• 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.

6
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

7
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
8
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
9
Just a few examples (of many, many applications)
Data mining
Production Planning
Staff Scheduling
Logistics
Cash Flow Planning
Project Mgt.
Market Research Planning
10
The Challenge of Staff Scheduling
1
So, how much staff is needed and how should they
by scheduled?
2
3
11
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
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

12
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
• 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

13
• Mathematical theory
• WWII
• Computers
• George Dantzig discovers Simplex Method
• Tons O research
• LINDO, CPLEX, GAMS, AMPL, GNU GLPK
• Terrific applications in industry
• Solver optimization for the masses
• Example Tactical Scheduling Analysis

14
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

15
Big Steps in Doing Optimization Modeling of
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
• relaxed constraints

Sensitivity Analysis
16
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
• 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)
from Lindo Systems, Inc.

17
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

18
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.)

19
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

20
• 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
21
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

22
• 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?
23
Pricing Model -- continued
• 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.

24
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
25
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.
26
Solver Main Dialog
Objective function cell a formula
Maximize, minimize or try to hit a target
Attempt to solve
Decision variables
See Options slide
Build constraints using
Clear dialog box
If you use range names, theyll show up here and
27
• 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
28
Solver Options
Solution search time limits
Save model to, or Load model from range on
Tell Solver problem is linear
Assume all decision variables gt0
29
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

30
Lets throw Solver at this function
31
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
32
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?

33
• 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?

34
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
35
Next Steps
• Lets solve another optimization problem together
• 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
• Well continue Excel based application
development using an optimization modeling example

36
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
37
• 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

38
• 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.

39
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
40
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
41