Optimization

Optimizationfrom Prof. Goldsmans lecture notes

Program Binusian 2012

OutlineWhat is a Model?

Operation Research

Optimization Example

Shortest Path

Introduction to Modeling


What is a Model?Abstraction, representation

Infinitely many models of the same reality

Often a model is created for a purposeA good model discards the irrelevantA good model retains what is crucial

Often we believe we understand something better after modeling it

We trust a model if it gives accurate predictions (qualitative or quantitative)


Note thatWhen you want to create a model

Collect the necessary dataDistinguish between the model input and model output Lengthy notes are worthlessKISS Rule


Operation ResearchHigher level of theoretical and mathematical orientation

Categorization of Operation Research:

Wants to model the problemEncounter the problems of estimating the values of the parametersParameters may not be constant over time

2 Approaches:Forget the fact that they are RV and use the model w/o RV taken into consideration Deterministic ApproachUse model w/ RV Probabilistic Approach


Optimization ExampleShortest Path of Auto Travel Routes

Distances are in miles


Shortest Path (Contd)Optimal Solution has length 270 miles

But it did not go to node d and a


Shortest Path (Contd)Algorithm actually finds a tree giving shortest paths from s to every node in graph


Shortest Path: DefinitionsGraph G= (V,E)V: vertex set, contains special vertices s and tE: edge set

Costs Cij on edges (i, j) in ECij >= 0no cycles with negative total cost

Cost of a path = sum of edge costsObjective: find min cost path from s to t


Shortest Path (Contd)A Mathematical ProblemAn Optimization Problem

It has: A set of possible solutions (paths from s to t)An objective function (min the sum of edge costs)

An algorithm that correctly and quickly solves cases of the shortest path problem, provided that

The instances satisfy Cij >= 0The instances are not too huge


ExampleGoal: use a car for 4 years at min cost


Purchase Cost

1st year

maint.

1 year

Resale

2nd year

maint

2 year

Resale

3rd year

maint

3rd year

Resale

New

Car

15000

1000

11000

1000

9000

1500

8000

Used

Car

5000

2000

4000

3000

3000

3500

2500

Example (Contd)Vertices of graph need not represent physical locationsV= {0,1,2,3,4}time 0, 1,...,4 in years

Seek least expensive path from 0 to 4

Edge cost from i to j: cost of buying a car at time i, using it, and selling it at time jfor each edge, pick cheapest alternative (new or used)


Example (Contd)Keep the new car for 1 year: 5000Keep the used car for 1 year: 3000Keep the new car for 2 years: 8000Keep the used car for 2 years:7000


Example (Contd)Continue and find the shortest path

Note that: shortest path does allow directed graph

i.e. you cannot go from node 2 to node 1 at a cost of 3000


Optimization Model Usage


What is A Successful Model?Models must fit the real problemRealism and Generality

Able to solve the model Solvability and Tractability


Modeling FAQTradeoff between realism and solvability

Good modelers know Different models limitationsFitting a model into a wider range of real problemsFitting a real problem into a model

Advanced modelers know how to Solve a wider range of modelsExtend the range of cases that can be solved with software tools


Optimization Models SpectrumNetworks LP Convex QP IP NLPShortest PathMin Span TreeMax FlowAssignmentTransportationMin Cost Flowportfoliooptimizationchemicalprocesses

materialsdesignblending

planninglogistics

schedulingproduction/distributionflow of materials


Mathematical ProgrammingLinear ProgrammingNonlinear ProgrammingInteger ProgrammingBinary ProgrammingQuadratic ProgrammingGeometric ProgrammingDynamic ProgrammingMixed Integer Programming


Mathematical Programming (Contd)Three important characteristics:

Decision VariablesObjective FunctionConstraints

Solving:GraphicallySimplex


A Piece of Cake ExampleFarmer Jones must determine how many acres of corn and wheat to plant this year. An acre of wheat yields 25 bushels of wheat and requires 10 hours of labor per week. an acre of corn yields 10 bushels of corn and requires 4 hours of labor per week. All wheat can be sold at $4 a bushel and all corn can be sold at $3 a bushel. Seven acres of land and 40 hours per week of labor are available. Government regulations require that at least 30 bushels of corn can be produced during the current year.

Formulate an LP whose solution will tell Farmer Jones how to maximize the total revenue from wheat and corn.

Find the solution using graphical method.


Assignment ProblemAssignment Problem

Simplest and easiestDef: there are N items or services available at N locations and N other locations that require one and only one of these N items or servicesExample: 4 types of product and 4 machines. Each machine can only produce one type of products.Objective: assign jobs to machines that will minimize the total cost


Assignment Problem (Contd)For unbalanced assignment problems

Add dummy rows or columnsSet the cost of these rows or columns to be all zeros


Solving Assignment ProblemUsing Hungarian Method Subtract the smallest number in each row from every entry in that rowSubtract the smallest number in each column from every entry in that columnA zero cost assignment (if it can be made) is optimal. If not, complete the Hungarian MethodDraw the min # of horizontal and vertical lines (no diagonals) that intersect all the zerosSubtract the smallest uncrossed number from every other uncrossed number and add it to all elements where the vertical and horizontal lines intersectIf a zero cost assignment can be made, it is optimalOw repeat steps 4 and 5 until one can be made


Solving Assignment Problem (Contd)Draw the min # of horizontal and vertical lines (no diagonals) that intersect all the zeros

Subtract the smallest uncrossed number from every other uncrossed number and add it to all elements where the vertical and horizontal lines intersectSmallest no = 1Row 4 & 5: Subtract 1Intersection: add 1

12345173200240111300011411130511160

12345173201240112300012400020500050


Solving Assignment Problem (Contd)If a zero cost assignment can be made optimal

Solution?4-1, 2-2, 3-3, 1-4, 5-5

12345173201240112300012400020500050


Hungarian Method: MaxObjective: maximize Note that max f(x) = min f(x)

Solve:Replace all the cost elements by their negativeContinue w/ the minimization procedure


Hungarian Method: Max (Contd)Replace all the cost elements by their negative

Subtract the smallest number in each row from every entry in that row (perform similar calculation for column)Find the smallest number per rowFind the smallest number per column

123415354221633546246314

12341-5-3-5-42-2-1-6-33-5-4-6-24-6-3-1-4

123410201245033120440352

123410000243023100340151


Transportation ProblemThere are M sources with something available and N destinations needing something

Difference:Assignment Problem: f(x): x yTransportation problem: f(x): x y1, y2,,ynf(x): x1, x2,,xm y


Transportation Problem (Contd)Mathematical Formulation:

ai = number of units available at ibj = #units needed at j cij = cost to ship 1 unit from i to jxij = #units shipped from source i to destination j


More ExampleSmallco, Inc.Smallco, Inc. manufactures two products, wooden toy cars and wooden toy trucks, made from mahogany. The profit margin for the two toys is $1.10 and $0.70, respectively. Based on careful market analysis, it appears that Smallco can sell about 2000 of each toy each week. However, Smallco can only obtain a limited amount of mahogany, roughly 3000 board-feet per week. Producing either toy requires 1 board foot of wood. What is the best mix of toys for Smallco to produce if the goal is to maximize total profit margin?


More Example (Contd)What is the objective?What can be controlled?How do the decisions affect the objective?What limits the decisions?The LP model is Solve it!


Example AgainDorial Auto manufactures luxury cars and trucks. The company believes that its most likely customers are high-income women and men. To reach these groups, Dorial Auto has embarked on an ambitious TV advertising campaign, and has decided to purchase 1-minute ad spots on two types of programs; comedy shows and football games. Each comedy commercial is seen by 7 million high-income women and 2 million high-income man. A 1-minute comedy spot costs $50,000 and a 1-minute football spot costs $100,000. Dorial wants to reach at least 28 million high-income women and 24 million high-income man. How should Dorian buy commercial time to reach their target at the lowest possible cost?


Model With Discrete VariablesModel with specific (discrete) values, is not a linear programming modelTypes:Integer ProgrammingMixed Integer ProgrammingBinary Integer Programming


Model With Discrete Variables (Contd)When do you need discrete variables?Continuous problemsyes/no aspects to the decisionHow many not How much

How to model it?Non-obvious trickTreat each linear function as a separate cost function, with its own variableAssociate a binary variable with each of the linear functionEnforce the requirement that you cannot do this if you dont have this (TRICK)


Model With Discrete Variables (Contd)It is trickyIt is not obvious and most of us (including myself) would have a hard time inventing this ourselvesCan be very difficult to solveExample: selecting among investment alternatives, designing supply chains, sourcing products, production/inventory planning, crew scheduling, vehicle routing, etc


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