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Optimization Lecture (2)

Optimization Lecture (2). 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 Lecture (2)

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  1. OptimizationLecture (2)

  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. Prototype example :The Acme Bicycle Company • The Acme Bicycle Company produces two kinds of bicycles: mountain bikes and street racers. Acme wishes to detremine the rate at which each type of bicycle should be produced in order to maximize the profits on the sales of the bicycles. Acme assumes that it can sell all of the bicycles produced. The physical data on the production process is available from the company engineer. • A different team produces each kind of bicycle and each team has a different maximum production rate : 2 mountain bikes per day and 3 racers per day, respectively. Producing a bicycle of either type requires the same amount of time on the metal finishing machine and this machine can process at most a total of 4 bicycles per day of either type. The company accountant estimates that mountain bikes are currently generating a profit of around 15 $ per bicycle, and racers a profit of around 10 $ per bicycle

  9. Solution • The first step in formulating the problem as a linear program is to identify the variables. • In this problem the variables are : • The production rates of mountain bikes (x1) • The production rates of racer bikes (x2) • There are bounds for these variables since X1 ≥ 0, x2 ≥ 0 non negative values of variables. 2. The second step is to write the objective function. Maximize Z = 15 x1+10 x2 ($/day) (maximize daily profit) 3. Use the variables to write the constrains: • mountain bike production limit x1≤ 2 bike/day • Racer bike production limit x2 ≤ 3 bike/day • Metal finishing machine x1+x2 ≤ 4 bike/day Production limit

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