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## Linear Optimization (Linear Programming)

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**Linear Optimization(Linear Programming)**Terry Turner, Lecturer School of Mathematical and Statistical Sciences Arizona State University**Absolute Optimization**• The domain is constrained to a closed and bounded region most of the time. Closed and bounded regions are guaranteed to have absolute extremes. • We analyze the partial derivatives for extremes within the region. • We analyze all paths for extremes. • We identify all corner points. • We evaluate everything we found. • The max is the highest z-value; • The min is the lowest z-value; • Discard anything in between.**The Basics**is a linear function, called A criterion function, or An objective function. is optimized subject to constraints: All constraints are linear. Any inequality form regularly seen in math works. and usually with real world situations.**Example**Maximize: Subject to:**Example**Find the maximum and minimum values for Subject to: Assume and represent the supervisors required to manage factories X and Y respectively.. The objective function tells the output based on your time spent managing each factory. Since X is newer it takes half as many supervisors as Y. Assume and represent the times required to manage factories X and Y respectively.. The objective function tells the output based on your time spent managing each factory. Since X is newer it takes half as much of your time as Y.**Example**A CEO invests some of his $120,000 savings in an account paying 5% annual interest and the rest in another account paying 6% annual interest. • He invests at least $60,000 in the account paying 6%. • The amount in the 6% account must be no more than three times the amount in the 5% account. How much must he invest in each account to maximize interest at the end of the year?**Linear Optimization**Terry Turner, Lecturer School of Mathematical and Statistical Sciences Arizona State University