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Lecture 8: Optimization with a Min objective

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)

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Lecture 8: Optimization with a Min objective

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

  2. 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)

  3. 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…

  4. 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.

  5. 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 offwith lower values of the objective variable *Constraints are lower limits --having higher requirement levels increases the minimum making us worse off

  6. 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…

  7. Most limiting requirementCereal A If we were only going to eat cereal A, we would have to eat 10 ounces to fulfill the Thiamine requirement.

  8. Most limiting requirementCereal B If we were only going to eat cereal B, we would have to eat 20 ounces to fulfill the Niacin requirement.

  9. 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…

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

  11. 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

  12. The Objective (Level Curve) • Price of Cereal A = 3.8 • Price of Cereal B = 4.2 • C = 3.8A + 4.2 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

  13. 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.

  14. 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.

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

  16. 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…

  17. 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?

  18. Algebraic Model Extended

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

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