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Lecture 18: TopicsInteger Program/Goal Program
– April 2
Assumptions 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
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
: Almost everyone enters into a management decision with multiple goals
Earn highest profits
What if these conflict?
Can we estimate the profit equivalent of a unit of sales?
Use no objective variable
Instead design a
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
Goal 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?
Example: Diet Problem
McDonalds Food Example
Total Cost (min)40.44Constraints (Slack/Bind) Calories5882.35Fat437.50Sodium12816.18Carbs539.12Fiber13.47Protein388.85Vitamin ABindingVitamin CBindingCalcium187.50Iron316.18
McD Dbl Cheese
McD Quarter Lber
McD Quarter Lber w/cheese
McD Dbl 1/4 Pounder
McD Big Mac
McD Big n' Tasty
McD Big n' Tasty w/ Cheese
McD Filet o' Fish
Diet 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.
Diet 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?
McDonalds won’t sell me 2.4 hamburgers
Do I buy 2 or 3?
Do I buy a Big Tasty or not?
Any number can be written as:
If a can be simplified to zero then we have an integer (nothing after the decimal)
In Solver: choose
Diet Problem w Goals and Integer Constraints
Instruct Solver to find integer values for all decision variables with integer (
Solve the same model as before adding only these constraints
How does it compare to rounding off the original solution?
Completely different situation
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
Rounding 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
Integer Programming Fact
Linear Program = VL
Rounded off linear program = VR
Integer Program = VI
For a min:
VL <= VI <= VR
For a max:
VL >= VI >= VR
Other 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
Integer Solution Algorithms
Allow us to relax the divisibility assumption of
Search program in the neighborhood of the relaxed solution
Active area in the fields of math programming, operations research, and applied mathematics
Lab and today’s lecture
Given a common set of food information and cost min objective
Student A solves linear program
Student B solves linear program with rounded solution