Optimizationsheet (1)

Recall

The optimization problem is as follows:

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) = 300 (equality constrain)Total no. of variables = 2 (x and y)No. of equality constraints = 1Independent variable: y since we can define x in terms in y As follows :Eliminate x using the equality constraint

12 8 96 300xy x y 204 8

12

yx

y

2. 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

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 functionMinimize f = 2100nA + 3600nB + 3780

nC (ton-mile/day)Constraints1. 10,000 nA + 20,000 nB + 23,000 nC ≤

600,000 ($)2. nA + 2nB + 2nC ≤145

(drivers)3. nA + nB + nC ≤ 30 (trucks)4. nA > 0 nB > 0 nC > 0

4 .

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

5.

solution