1. Optimization sheet (1)

2. Recall
The optimization problem is as follows:

3. Examples
For each of the following problems, formulate the objective function, the equality constraints (if any), and the inequality constraints (if any). Specify and list the independent variables.
1. A poster is to contain 300 cm2 of printed matter with margins of 6 cm at the top and bottom and 4 cm at each side. Find the overall dimensions that minimize the total area of the poster.
Solution:
Minimize: f (x,y) = xy (objective function)
Subject to: (x-8) (y-12) = (equality constrain)
Total no. of variables = 2 (x and y)
No. of equality constraints = 1
Independent variable: y since we can define x in terms in y
As follows :
Eliminate x using the equality constraint

4. Find the area of the largest rectangle with its lower base on the x axis and whose corners are bounded at the top by the curve y = 10 – x2.
solution
Maximize: A = b * h (objective function)
Subject to: h = 10-(b/2)2 (equality constrains)
and Total no. of variables = 2 (b, h)
No. of equality constraints = 1
Independent variable: b since we can define h
In terms of b

5. Solution: Objective function Constraints
3. A trucking company has borrowed $600,000 for new equipment and is contemplating three kinds of trucks. Truck A costs $10,000, truck B $20,000, and truck C $23,000. How many trucks of each kind should be ordered to obtain the greatest capacity in ton-miles per day based on the following data?
Truck A requires one driver per day and produces 2100 ton-miles per day.
Truck B requires two drivers per day and produces 3600 ton-miles per day.
Truck C requires two drivers per day and produces 3780 ton-miles per day.
There is a limit of 30 trucks and 145 drivers.
Formulate a complete mathematical statement of the problem, and label each individual part, identifying the objective function and constraints with the correct units ($, days, etc.). Make a list of the variables by names and symbol plus units. Do not solve.
Solution:
Let nA = no. of trucks of type A
nB= no. of trucks of type B
nC = no. of trucks of type C
Need to define the problem to get the greatest life cycle for the trucks in terms of ton-mile/day, this means to minimize the tons that covered by this truck /day so the
Objective function
Minimize f = 2100nA nB nC (ton-mile/day)
Constraints
1. 10,000 nA + 20,000 nB + 23,000 nC ≤ 600,000 ($)
2. nA + 2nB + 2nC ≤145 (drivers)
nA + nB + nC ≤ 30 (trucks)
nA > 0 nB > 0 nC > 0

6. 4.

7. The variables are: A, B, T, t
solution
The variables are: A, B, T, t
A, B depend on the time and temperature but T does not depend on A nor B or t.
The independent variable is T
The dependent variable are A, B
The equality constrains are the 4 given constrains.
The inequality constrains is T≤ 282 0F
Also T≥ 0, A ≥ 0, B ≥ o and t ≥ 0

8. 5.

9. solution