1 / 9

Optimization sheet (1)

Optimization sheet (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.

tracen
Download Presentation

Optimization sheet (1)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Optimizationsheet (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) = 300 (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: bsince we can define h In terms of b

  5. 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 + 3600nB + 3780 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. 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

More Related