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 Example XTBTCNBMBMCDecision More X More X 3* More X Less X Less X Less X Note: X is the activity or choice variable and orange values were originally not in Table. #3, page 139

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 MB>MC and decrease activity if MB MC.

4.10 Unconstrained Maximization Continuous 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 #2, page 143

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 Calculus Unconstrained Max  B(x)=170x-x 2 and C(x)=100-10x+2x 2 a. MB = B ‘ (x)=170-2x MC = C ‘ (x)=-10+4x b. Profit max requires MB=MC and 170-2x = -10+4x or 6x=180 or x=30 or NB = B –C = 170x-x 2 – (100-10x+2x 2 ) NB = -3x x-100 NB ‘ = -6x+180 = 0 or x = 30 c. NB(30) = -3(30) (30)-100 NB(30) = 2600 #1, page 149

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 #11, page 141 PersonSales/DayWage/DayMB/$1 Jane600$2003 Joe450$1503 Joan400$1004

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. MB A /P A = MB B /P B = MB C /P C

4.17 Example of Constrained Max #7, page 145 C ServedMB/P in $000 Order of hire Exp $000 empHSBSHSBSHSBSTotal a b Constraint is satisfied MB/P are equal for both when HS=5 And BS=2