Lecture 18: Topics Integer Program/Goal Program AGEC 352 Spring 2011 – April 6 R. Keeney

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

Optimization 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?

If you can’t design the tradeoff, you might be able to prioritize… Priority of goals for my college time ◦ 1) Find a spouse ◦ 2) Find a spouse that will get a good job after graduation ◦ 3) Get a degree that will get me a good job ◦ 4) Get good grades to keep my folks from complaining ◦ 5) Learn stuff

Goal 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

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)40.44 Constraints (Slack/Bind) Calories Fat Sodium Carbs Fiber13.47 Protein Vitamin ABinding Vitamin CBinding Calcium Iron Foods (quantity) McD Hamburger0.00 McD Cheeseburger0.00 McD Dbl Cheese3.68 McD Quarter Lber0.00 McD Quarter Lber w/cheese0.00 McD Dbl 1/4 Pounder0.00 McD Big Mac0.00 McD Big n' Tasty14.71 McD Big n' Tasty w/ Cheese0.00 McD Filet o' Fish0.00 McD McChicken0.00

Diet Problem as a Goal Program Units are a problem (as always) Convert to percentages Set a maximum cost = $10 ◦ Eat ten Double Cheeseburgers Total Cost (min)10.00 Constraints (Slack/Short) Calories Fat Sodium Carbs Fiber Protein Vitamin A Vitamin C Calcium Iron75.00 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 Program With 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 = (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? 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

Diet 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?

Completely different situation 1 Cheeseburger 1 Filet o’ fish Total loss = (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 = / Total Cost = 1.60

Integer 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

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 linear programming Search program in the neighborhood of the relaxed solution ◦ Active area in the fields of math programming, operations research, and applied mathematics

Quiz on Monday 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?