4.1

The Theory of Optimization

Optimizing Theory deals with the task of finding the “best” outcome or alternative– Maximums and– Minimums

What output will lead to maximum profit? What combination of inputs will lead to

minimum cost of producing a given level of output?

4.2

Concepts and Terminology

The objective function is the function to be maximized or minimized. In many cases, our goal is to maximize profits. Therefore, the profit function would be the objective function.

4.3

Concepts and Terminology

Activities or choice variables are the variables controlled by the decision-maker that influence the value of the objective function.

For example, the manager can change the level of output and influence the level of profits.

Choice variables can be continuous or discrete.

4.4

Concepts and Terminology

Optimization problems can be constrained or unconstrained.– Unconstrained optimization problems allow the

decision-maker to choose from an unrestricted set of values for the activity or choice variable.

– Constrained optimization problems allow the decision-maker to choose from a restricted set of values for the activity or choice variable.

4.5

Concepts and Terminology

For example a constrained maximization problem occurs if a manager has a fixed budget(constraint since it limits how much of the various inputs that can be purchased)and desires to maximize output.

Another example is decision-maker desires to minimize cost of producing a given level of output.

4.6

Concepts and Terminology

Marginal analysis is a powerful tool that enables decision-maker’s to solve optimization problems.

Marginal analysis focuses on the change in the objective function in response to small changes in the choice variable(s).

End product will be two rules – one for unconstrained and another for constrained optimization problems.

4.7

Unconstrained Maximization

Decision-maker chooses the value of the activity so as to maximize the net benefit.

Net Benefit = Total Benefit – Total Cost

NB = TB – TC An example Profit = TR – TC The level of the activity that leads to

maximum net benefit is the optimal level.

4.8

An ExampleX TB TC NB MB MC Decision

0 0 0 0

1 35 8 27 35 8 More X

2 65 18 47 30 10 More X

3*3* 85 30 55 20 12 More X

4 95 44 51 10 14 Less X

5 103 60 43 8 16 Less X

6 108 80 28 5 20 Less X

Note: X is the activity or choice variable and orange values were originally not in Table.

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4.9

Unconstrained Maximization Discrete Activity Variable

Note in the previous example, we assumed that the choice variable or activity was discrete – thus, the optimal solution was X=3.

The Principle is increase activity if The Principle is increase activity if MB>MC and decrease activity if MB<MC. MB>MC and decrease activity if MB<MC. Optimal solution is the last activity value Optimal solution is the last activity value for which MB>MC.for which MB>MC.

4.10

Unconstrained MaximizationContinuous Activity Variable

To maximize net benefit you increase the activity if MB>MC and decrease the activity if MB<MC. The Optimal solution occurs when MB=MC.

Note MB generally declines and MC tends to rise with increases in the activity. At the point they intersect is the optimal point.

4.11

An Example

Suppose MC = 40P and MR = 1,000 – 10P What is the profit maximizing level of P?

Answer: Profits are max where MR=MC 40P =1,000-10P or 50P = 1,000 and P =20

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4.12

Using Calculus for Unconstrained Max

Remember derivatives are essentially slopes or marginals. Thus MB and MC are the derivatives of the TB and TC functions respectively.

Also maximum NB requires that the slope or derivative of the NB function is zero.

4.13

An Example Using CalculusUnconstrained Max

B(x)=170x-x2 and C(x)=100-10x+2x2

a. MB = B‘(x)=170-2xMC = C‘(x)=-10+4x

b. Profit max requires MB=MC and170-2x = -10+4x or 6x=180 or x=30or NB = B –C = 170x-x2 – (100-10x+2x2)NB = -3x2+180x-100NB‘ = -6x+180 = 0 or x = 30

c. NB(30) = -3(30)2+180(30)-100NB(30) = 2600

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4.14

Sunk and Fixed Costs

Sunk costs are costs that have been previously paid and are unrecoverable.

Fixed costs are costs that must be paid(in the short run) no matter what level of the activity is chosen.

The important principle is that these costs are irrelevant for short run decisions. Only costs that change(variable costs) are relevant.

4.15

Marginal Benefit per Dollar

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Person Sales/Day Wage/Day MB/$1

Jane 600 $200 3

Joe 450 $150 3

Joan 400 $100 4

4.16

Constrained Max

In order to maximize total benefits subject to a constraint on the level of activities, employ each activity to the point at which the marginal benefit to price ratio is equal for all activities. Same is true for constrained minimization.

MBA/PA = MBB/PB = MBC/PC

4.17

Example of Constrained Max

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C Served MB/P in $000

Order of hire

Exp

$000

emp HS BS HS BS HS BS Total

1 120 100 6 3.33 1 5 50

2 220 190 5 3 2 6a 100

3 300 270 4 2.67 3 120

4 370 330 3.5 2 4 140

5 430 380 3 1.67 6b 160

6 470 410 2 1

Constraint is satisfiedMB/P are equal for both when HS=5And BS=2