Lecture 8: Optimization with a Min objective AGEC 352 Spring 2012 – February 8 R. Keeney

Upcoming Today: Minimization lecture Next Week (Week 6) ◦ 2/13: Quiz on lecture 4-8; review HW 3 ◦ 2/14: Lab on Minimization ◦ 2/15: Sensitivity Week 7 ◦ 2/20: Exam Review in Class (Assignment) ◦ 2/21: No lab, Office Hrs from 9-12 ◦ 2/22: Exam I (50 mins., 30 ?’s, MC & T/F)

Profits as the objective Profit has been used so far as our objective variable and we assumed that this variable should be maximized What if our problem has no revenue? ◦ Max P = R – C  R = 0 then  Max P = -C We could maximize profits by finding the maximum of the negative of costs. Or…

Just treat Cost as the objective variable Max –C Min C ◦ These two are equivalent. Any minimization problem can be rewritten as a maximization of the negative of the objective variable. Given this relationship, we should expect everything about a basic minimization problem to be opposite (negative) of our basic maximization problem.

Comparison of 2 variable problems *Objective = max --better off with higher values of the objective variable *Constraints are upper limits --having higher resource levels increases the maximum making us better off *Objective = min --better off with lower values of the objective variable *Constraints are lower limits --having higher requirement levels increases the minimum making us worse off

Constraint Analysis We found the ‘most limiting resource’ in maximization problems ◦ Divide RHS by coefficient for an activity and choose the smallest Now we can find the ‘most limiting requirement’ ◦ Divide the RHS by coefficient for an activity and choose the largest Graphically…

Most limiting requirement Cereal A RequirementSupplied by Cereal A Total required Total/ required Thiamine Niacin Calories If we were only going to eat cereal A, we would have to eat 10 ounces to fulfill the Thiamine requirement.

Most limiting requirement Cereal B RequirementSupplied by Cereal B Total required Total/ required Thiamine Niacin Calories If we were only going to eat cereal B, we would have to eat 20 ounces to fulfill the Niacin requirement.

Some things we know just from the constraints Feasible space graph ◦ Cereal A axis intersects at A = 10 ◦ Cereal B axis intersects at B = 20 The cereals have different most limiting requirements, a mixture of the two cereals is very likely to be cheaper than eating only one Calories is not the most limiting resource for either food. It should not bind in the optimal solution…

Feasible space of a maximization problem (see lecture for Jan. 25) Feasible space is inside of the lines.

Feasible space of the cereal mix minimization problem *Feasible space is everything northeast of the lines. *Completely defined by Thiamine and Niacin. *Three corner points: A=0 & B=20 A=10 & B=0 A=4.4 & B=2.2

The Objective (Level Curve) Price of Cereal A = 3.8 Price of Cereal B = 4.2 C = 3.8A B ◦ Rewrite this isolating B (the y-axis variable) B = C/4.2 – (3.8/4.2)A ◦ Cost measured in units of Cereal B  Intercept is determined by C, so we want the lowest intercept  Slope converts A to units of B

Model with Level Curves *Lower intercepts indicate lower costs *Objective is improving when we lower the intercept due to minimization *Cost = 20 is not feasible (no point gives both enough thiamine and niacin) *Cost = 30 is feasible but is not optimal.

Back to economics This is an expenditure minimization model ◦ Similar to the utility problem faced by consumers. ◦ What is the cheapest way to reach a target level of satisfaction. ◦ Indifference curves and an isoexpenditure line… Also the input-input model of producer problems that use isoquants.

Economic Model X1 X2 U = target level *Expenditures decrease in this direction but must be large enough to meet the target utility. *Optimum is the tangency point.

Basic versus Non-basic models Basic models have only one type of constraint ◦ Maximization  <= constraints ◦ Minimization  => constraints More realistic models will have more options and may have both types of constraints…

Calories in the cereal problem What if we cared about both ◦ Eating enough calories and ◦ Not eating too many calories How would we model this?

Algebraic Model Extended

New Feasible Space Feasible Space is the triangle between Niacin, Thiamine, and Max Cal constraints.