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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.

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Optimization sheet (1)

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  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

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  9. solution

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