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Sec 4.5

Sec 4.5. Optimization (Applied Max/Min Problems). Optimization Problems. Given: A task. Objective: Find the “ best ” way to complete the task. Depends on our concern. Specific Examples:. (1) Cheapest way to do a certain thing;. (2) Fastest way to do a certain thing;.

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Sec 4.5

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  1. Sec 4.5 Optimization (Applied Max/Min Problems)

  2. Optimization Problems Given: A task. Objective: Find the “best” way to complete the task. Depends on our concern. Specific Examples: (1) Cheapest way to do a certain thing; (2) Fastest way to do a certain thing; (3) To obtain the largest among a group of things…

  3. Example (1) Manufacturing a can Task: To manufacture a (cylindrical) can with a given capacity, say 500 mL. “Best”: We want to use the least amount of material.

  4. Example (2) Making a window Task: To manufacture a window with a given shape and perimeter. “Best”: We want the window to admit the greatest possible amount of light. Note: Extra (specific) information will be provided …

  5. Example (3) Fitting a rectangle inside a region Task: To place a rectangle inside a given region. “Best”: We want the rectangle to be as large as possible (in terms of area).

  6. Example (4) Finding a ladder Task: To find a ladder to reach over a fence to a building. “Best”: We want the shortest ladder that can do the job.

  7. Example (5) Offshore oil-well problem Task: To build pipes for oil transportation. “Best”: We want the cost to be as low as possible.

  8. Example (6) Rescue Operation Task: To get to a plane crash site. “Best”: We want to get there as quickly as possible.

  9. Example (1) Manufacturing a can Task: To manufacture a (cylindrical) can with a given capacity, say 500 mL. “Best”: We want to use the least amount of material. Height h Such cans come in all shapes and sizes. They differ from one another in dimensions. Base radius r To find the absolute minimum of Amount of material,A.

  10. Example (2) Making a window Task: To manufacture a window with a given shape and perimeter. “Best”: We want the window to admit the greatest possible amount of light. Extra information: The perimeter, say 4 m. Such windows come in all sizes. They differ from one another in dimensions. r h To find the absolute maximum of Area of the window,A.

  11. Method of Solution • Read the question carefully, understand the task involved, and the many possible ways of completing the task. • Identify the quantity that you want to maximize (or minimize) while completing the task. • Express this quantity as a function of some suitably chosen parameter that represents the possible ways of completing the task. • Find the absolute maximum (or minimum) of your quantity.

  12. Example (3) Fitting a rectangle inside a region Task: To place a rectangle inside a given region. “Best”: We want the rectangle to be as large as possible (in terms of area). Extra information: The specifics about the region, say … y = 8 – x2 Such rectangles come in all shapes and sizes. (x,y) They differ from one another in dimensions … … or, the location of their corners. y = 0 To find the absolute maximum of Area of the rectangle,A.

  13. Example (4) Finding a ladder Task: To find a ladder to reach over a fence to a building. “Best”: We want the shortest ladder that can do the job. Extra information: The specifics about the fence, say … Such ladders come in all sizes. They differ from one another in inclination … L … or, the angles they make with the ground. 2 m 2 m To find the absolute minimum of Length of the ladder, L.

  14. Example (5) Offshore oil-well problem Task: To build pipes for oil transportation. “Best”: We want the cost to be as low as possible. Such pipes come in many forms. To find the absolute minimum of cost, C. Extra information: The specifics about various costs, say … They differ from one another in where the junction is located … Underwater pipe costs $ 400,000 per km Pipe on land costs $ 300,000 per km Location of oil well: 3 km off shore 3 km Location of factory: 6 km down x km 6 km

  15. Example (6) Rescue Operation Task: To get to a plane crash site. “Best”: We want to get there as quickly as possible. Many paths lead to the plane. To find the absolute minimum of time, T. Extra information: The specifics about speeds, say … They differ from one another in where the “take-off point” is located … Speed on sand 50 km/h 5 km Speed on land 130 km/h Location of plane: 5 km off road “Take-off point” Location of hospital: 8 km down x km 8 km

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