Linear Programming Examples with Solutions

Linear Programming Examples with Solutions - UNC-Wilmington...

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UNC-Wilmington ECN 321 Department of Economics and Finance Dr. Chris Dumas Examples of Consumer Choice Problems that can be solved with Linear Programming Example (1): Alicia Silverstone's Problem Suppose Cher (played by Alicia Silverstone in the movie, Clueless) is trying to decide how to allocate her leisure time on a particular vacation weekday (you've got to see the movie, but, suffice it to say that, for Cher, every weekday is a vacation day). Specifically, she is trying to decide how many hours to spend at the beach and how many hours to spend at the mall. All else equal, she figures that an hour spent at the beach gives her about one-third more satisfaction than an hour spent at the mall. At most, suppose she has 8 hours of leisure time to divide between the beach and the mall. However, due to the weather, suppose there will be only 6 good beach hours this day. Also, due to hours of operation, suppose the mall is open only 10 hours. In addition, suppose Cher spends, on average, $5 per hour at the beach and $10 per hour at the mall. Finally, suppose Cher has a total of $60 to spend on leisure this day. What will Cher choose to do? Solution: Identify the choice variables: Let x1 = beach hours and x2 = mall hours. Set up the optimization problem: max U = 1.33 x1 + 1 x2 x1, x2 Subject to: (1) x1 0 (2) x2 0 (3) x1 + x2 8 (4) x1 6 (5) x2 10 (6) 5 x1 + 10 x2 60 To facilitate graphing, constraint (3) above can be re-written as : x2 8 - x1 Similarly, constraint (6) above can be re-written as : 10 x2 60 - 5 x1 x2 (60 - 5 x1)/10 x2 6 - 1/2 x1 1
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UNC-Wilmington ECN 321 Department of Economics and Finance Dr. Chris Dumas To find the Feasible Region, graph the constraints: Finding the coordinates for Extreme Point C: The x2's of constraints (3) and (6) are equal at Point C; hence, 8 - x1 = 6 - 1/2 x1 2 = 1/2 x1 x1 = 4 Plugging x1 = 4 back into either constraint, we find x2 = 4 Thus, Extreme Point C is (4,4). Finding the coordinates for Extreme Point D:
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Linear Programming Examples with Solutions - UNC-Wilmington...

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