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Lecture 18: TopicsInteger Program/Goal Program

AGEC 352

Spring 2012

– April 2

R. Keeney

Slide2Assumptions of Classical Linear Programming

There are numerous assumptions that are in place when you solve an LP

Proportionality – straight line behavior

Optimization – choose a single measure and make it best

Divisibility – agents can choose any real number for decision variables

We will break the first one next week

We will break 2 and 3 today

Slide3Optimization

A single indicator variable (often measured in dollars) is used to determine well-being of the decision maker

The decision maker wants to maximize or minimize this indicator

Tradeoff

: Almost everyone enters into a management decision with multiple goals

Earn highest profits

Maximize sales

What if these conflict?

Can we estimate the profit equivalent of a unit of sales?

Slide4Goal Programming

Use no objective variable

Instead design a

loss variable

that keeps track of how far short of a number of goals you fall

Forces you to set targets for a number of objectives

You may need to establish the tradeoff rates for each of these

Sales vs. Profits

Target Sales = $1,000,000

Target Profits = $125,000

Slide5Goal Programming for Competing Objectives

Objective variable = V

V = (125,000 – Profits) + (1,000,000 – Sales)

Minimize this variable

What does it mean if:

V < 0?

V > 0?

V = 0?

Do the units make sense?

What if profits are twice as important as sales?

Slide6Example: Diet Problem

McDonalds Food Example

Total Cost (min)40.44Constraints (Slack/Bind) Calories5882.35Fat437.50Sodium12816.18Carbs539.12Fiber13.47Protein388.85Vitamin ABindingVitamin CBindingCalcium187.50Iron316.18

Foods (quantity)

McD

Hamburger

0.00

McD

Cheeseburger

0.00

McD Dbl Cheese

3.68

McD Quarter Lber

0.00

McD Quarter Lber w/cheese

0.00

McD Dbl 1/4 Pounder

0.00

McD Big Mac

0.00

McD Big n' Tasty

14.71

McD Big n' Tasty w/ Cheese

0.00

McD Filet o' Fish

0.00

McD McChicken

0.00

Slide7Diet Problem as a Goal Program

Units are a problem (as always)Convert to percentagesSet a maximum cost = $10Eat ten Double Cheeseburgers

Total Cost (min)10.00Constraints (Slack/Short) Calories1900.00Fat220.00Sodium9500.00Carbs210.00Fiber-18.00Protein194.00Vitamin A-25.00Vitamin C-105.00Calcium125.00Iron75.00

This is not a desirable result. Over consume some nutrients to change the objective.

Need to add caps to all of the nutrients.

Slide8Diet Problem as Goal ProgramWith Nutrient Caps

No excess of any nutrient

Minimize the percentage loss of our target daily nutrition needs

Spend at most $10

Eat 2.4 Hamburgers and 0.35 Big Tasty

Cost is $2.80

Total Loss = 549.82 (max is 1000)

Hit the target for daily fat (30gm) and sodium (1500mg) still need everything else

What do we learn from this model?

Anyone see an additional problem?

Slide9McDonalds won’t sell me 2.4 hamburgers

Do I buy 2 or 3?

Do I buy a Big Tasty or not?

Integer constraints

Any number can be written as:

X+a

/b

If a can be simplified to zero then we have an integer (nothing after the decimal)

In Solver: choose

int

Slide10Diet Problem w Goals and Integer Constraints

Instruct Solver to find integer values for all decision variables with integer (

int

) constraints

Solve the same model as before adding only these constraints

How does it compare to rounding off the original solution?

Slide11Completely different situation

1 Cheeseburger

1 Filet o’ fish

Total loss = 601.05 (max 1000)

Total cost = $2.94

We are worse off because McDonald’s will not sell us parts of a hamburger and Big N Tasty

Paying more for less nutrition

Slide12Rounding to an integer solution

In many instances analysts solve the relaxed integer program and just round the solution

When is that appropriate?

In this case:

Round down (otherwise violate the boundaries)

2 hamburgers, no Big Tasty

Total Loss = 703.97 / Total Cost = 1.60

Slide13Integer Programming Fact

Objective variable

Linear Program = VL

Rounded off linear program = VR

Integer Program = VI

For a min:

VL <= VI <= VR

For a max:

VL >= VI >= VR

Slide14Other issues in integer programming

No sensitivity analysis or shadow prices are calculated

Can’t find them via the simplex or calculus methods

Have to resolve the model with a one unit change to the RHS of the constraint you are interested in

Complex mathematics to solve

Large models (e.g. Sudoku) can take a long time to solve because many combinations must be checked

Slide15Integer Solution Algorithms

Allow us to relax the divisibility assumption of

linear programming

Search program in the neighborhood of the relaxed solution

Active area in the fields of math programming, operations research, and applied mathematics

Slide16Lab/Project

Diet problem

Lab and today’s lecture

Example question:

Given a common set of food information and cost min objective

Student A solves linear program

Student B solves linear program with rounded solution

Student C solves integer program

Who has the lowest objective variable?

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