Optimization Techniques

1
Optimization Techniques
2
Objectives of Discussion

• Present concepts terminology used in discussing
  the optimization process
• Demonstrate use of marginal analysis in the
  optimization process
• Present principles for single variable
  optimization processes
• Present principles for multiple variable
  optimization processes
• Develop principles for constrained optimization
  processes

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Some Basic Terms

• Objective function
• Expresses relationship between the outcome
  variable that is to be optimized the decision
  variables
• Choice or decision variables
• Variables that decision maker can manipulate to
  alter value of objective function
• Can be either discrete, or continuous
• Unconstrained vs. constrained optimization
• Unconstrained--unrestricted set of decision
  variables
• In reality, most problems involve constraints

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The Meaning of Optimization

• Choose course of action that gives the most
  desirable outcome
• Where the course of action involves choosing
  values for one, or more, decision variables
  e.g.
• Quantities of various products to produce
• Quantities of various resources to use in
  production
• Desirable outcomes are usually measured in
  terms of such things as
• Level of profits, costs, revenue, etc. for firms
• Level of satisfaction, expenditures, wages, etc.
  for households

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An Overview of Optimizing Techniques

• Appropriate technique depends on
• Number of decision variables
• Whether decision variable(s) are discrete or
  continuous
• Presence of constraints
• Preliminary steps in optimization
• Identify the outcome measure
• Identify the decision variable(s)
• Identify relationships between decision variables
  and Marginal Benefits Marginal Costs
• Determine if constraint(s) exist how they are
  related to decision variables

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Irrelevance of Sunk, Fixed, Average Costs

• Sunk costs
• Previously paid cannot be recovered
• Fixed costs
• Constant must be paid no matter the level of
  activity
• Average (or unit) costs
• Computed by dividing total cost by the number of
  units of the activity
• These costs do not affect marginal cost are
  irrelevant for optimal decisions

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Optimal Level of Activity
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Marginal Analysis-Unconstrained

• In any optimization problem, objective can be
  expressed as difference between a total benefit
  total cost function
• Focus is on the change in the level of benefits
  costs associated with small changes in a decision
  variable
• Marginal Benefit Change in Total Benefit for a
  small change in value of decision variable
• Marginal Cost Change in Total Cost for a small
  change in value of decision variable
• Optimizing rules
• Choose value for decision variable that equates
  (to the extent possible) MB MC
• If MB gt MC, increase level of decision variable
• If MC gt MB, decrease level of decision variable

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Relating Marginals to Totals
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Example- Unconstrained, Single Var.

• Find profit max. Q when TR 100Q - .6Q2 and
  TC 2100 - 9Q .6Q2
• ? TR - TC 100Q - .6Q2 - 2100 - 9Q .6Q2
• Find MR MC functions
• MR d(TR)/dQ 100 - 1.2Q
• MC d(TC)/dQ -9 1.2Q
• Set MR MC and solve for Q
• 100 - 1.2Q -9 1.2Q
• -2.4Q -109
• Q 45.42
• Substitute for Q and find TR, TC ?
• TR 100(45.42) - .6(45.42)2 3304.21
• TC 2100 - 9(45.42) .6(45.42)2 2929.00
• ? TR - TC 3304.21 - 2929 375.20

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Unconstrained-Two Variable

• Same principle as in one-variable with a
  qualification
• Choose values of each decision variable so that
  MB MC
• Qualification When working with one decision
  variable, must find a way to hold other decision
  variables constant
• Do this through use of partial derivatives--see
  Appendix

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Example--Allocating the Advertising Budget

• Firm uses two advertising media, TV Newspaper
• Costs per unit of advertising TV 10, NP 2
• Firms TR TC are affected as follows
• TR 20T 5N - T2 - 5N2
• TC 10T 2N
• How many units of each should firm use?
• Find MR MC for each media and set them equal
• Take partial derivatives of TR TC w.r.t. T and
  N
• MRT 20 - 2T MCT 10 set MRT MCT
• Solve for T 5
• MRN 5 - N MCN 2 set MRN MCN
• Solve for N 3
• ? TR - TC 20(5) 5(3) - (5)2 - .5(3)2 -
  10(5) - 2(3) 29.5

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Constrained Optimization

• Still use marginal analysis, but instead of
  focusing on MB MC, we focus on ratio of MB to
  MC
• Optimizing rule
• Choose levels of decision variables so that ratio
  of MB to MC is equal for all decision variables,
  i.e.
• MBa/MCa MBb/MCb . MBn/MCn
• The ratios indicate the contribution to
  benefits per of cost outlay for a small change
  in the decision variables

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Extending the Adv. Example

• In previous example when T lt 5,
• MRT/MCT gt 1 and increasing T adds more to TR than
  to TC
• For example, let T 4
• MRT 20 - 2(4) 12 MCT 10
• ? MRT/MCT 1.2 which means that a one unit
  increase in T adds 1.20 to revenue for each
  1.00 added to cost
• Also now assume that N 2
• MRN 5 - 2 3 MCN 2
• ? MRN/MCN 1.5 which means that a one unit
  increase in N adds 1.50 to revenue for each
  1.00 added to cost

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Imposing a Budget Constraint

• Suppose that the advertising budget in preceding
  example is limited to 50
• At previous unit prices of PT 10 PN 2,
  when T 4 and N 2, we would have
  spent 44
• Assuming that we can buy fractions of ads, how
  would we spend the remaining 6
• Answer allocate limited funds to get largest
  increase in revenue per of cost outlay--? first
  allocate money to N
• Note that as we allocate money the ratios change

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Steps in Optimal Solution

• Set ratios equal to each other and find optimal
  relationship between T and N
• MRT/MCT MRN/MCN
• (20-2T)/10 (5-N)/2 cross multiply simplify
• T 2.5N -2.5
• Substitute the above into budget constraint
  determine amount of T N to be purchased
• Budget constraint 50 10T 2N
• Substituting for T 50 102.5N - 2.5 2N
• Solve for N N 2.778
• Solve for T T 2.5(2.778) -2.5 4.44