Chapter 7 b625aswc07.pptx - Introduction to Linear...

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Introduction to Linear Programming
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Linear? To get a feel for what linear means let’s think about a simple example. You have $10 to spend and you can buy either M&M’s at 50 cents a bag or Celery at $1 per stalk. Let’s create two columns of numbers where each row will show a combination of Celery and M&M’s you can buy and you will spend the whole $10. The next slide has the information.
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example So, if you buy no celery you can buy 20 bags of M&M’s. If you buy no M&M’s you can have 10 celery stalks. Check the other combinations. The linear idea is related to the idea that when the amount of celery you buy changes by 1 stalk the M&M’s always changes by 2 bags (other examples may not be ratio of 1 to 2). Let’s see on the next screen how we can put this information into a graph, OK :) Celery M&M’s 0 20 1 18 2 16 3 14 4 12 5 10 6 8 7 6 8 4 9 2 10 0
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example M&M’s M Celery C (10, 0) (0, 20) (5, 10) We have to decide which “axis” M&M’s will take and which Celery will take. Let’s say the horizontal axis has been called C and the vertical axis M. I put M&M’s on the vertical axis. In parentheses you see points (C, M). This means start at the origin, then go over the value C and then go up the value M. The origin (0, 0) All the points are on a straight line .
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example In the graph we drew, the points on the line imply we spend the whole $10 in the example. If we did not spend the whole $10 we would be inside (south and west) the line. But, with only $10 we can not be outside (north and east) the line. In general, the line represents a limit to what we can get (note the origin is our reference point). In the current example it represents an upper limit. We will see down the road an example where the line might be a lower limit.
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equation If I = the consumer income in dollars Pc = the price per unit of Celery Pm = the price per unit of M&M’s C = the amount of celery you buy M = the amount of M&M’s you buy, then, in general, the amount you buy is I = (Pc)(C) + (Pm)(M) Now, we have I = $10, Pc = $1 and Pm = 0.50 so the equation becomes $10 = $1(C) + $0.05(M). Since we can have various combinations of C and M, C and M are left general in the equation. Once we decide on the amount of C, the amount of M we want must satisfy the equation (if we are on the line), and vice versa.
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Typical linear programming problem Linear programming is a way to handle certain types of optimization problems.
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