# Linear programming with absolute value objective function

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 P: 76 1. The problem statement, all variables and given/known data Minimize |2x1-3x2| subject to x1+x2≤5 -x1+x2≥-1 x1≥0, x2≥0 (a) Solve the problem graphically. (b) Formulate a linear program that could be used to solve the problem. Use software to solve your LP and show how to reconstruct a solution to the original problem. 2. Relevant equations 3. The attempt at a solution I started graphing the problem and posted an attachment of what I have. I usually solve these by way of the corner point method (just to check that my graphical method is correct), but once I found my corner points for this problem I found that 2 of them have the same value but are not adjacent CPF solutions (0,0)=0 (1,0)=1 (3,2)=0 (0,5)=15 This is what's confusing me... also, how would I show the graphical method when absolute value is involved? I started to try it, but I would like to know if I'm doing it correctly before going further. Attached Thumbnails
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P: 5,186
 Quote by csc2iffy 1. The problem statement, all variables and given/known data Minimize |2x1-3x2| subject to x1+x2≤5 -x1+x2≥-1 x1≥0, x2≥0 (a) Solve the problem graphically. (b) Formulate a linear program that could be used to solve the problem. Use software to solve your LP and show how to reconstruct a solution to the original problem. 2. Relevant equations 3. The attempt at a solution I started graphing the problem and posted an attachment of what I have. I usually solve these by way of the corner point method (just to check that my graphical method is correct), but once I found my corner points for this problem I found that 2 of them have the same value but are not adjacent CPF solutions (0,0)=0 (1,0)=1 (3,2)=0 (0,5)=15 This is what's confusing me... also, how would I show the graphical method when absolute value is involved? I started to try it, but I would like to know if I'm doing it correctly before going further.
I assume the question wants you to give an LP formulation---that is, a formulation that does *not* have any absolute values in it. Formulating such problems is standard, and can be found in any introductory LP or Operations Research book. You can even look on line.

RGV
 P: 76 Yes that's basically what I'm trying to ask about here. I tried looking in my book but I can't seem to find absolute value objective functions. I remember my teacher talking about this but her example isn't about absolute value objective let xi = xi+ - xi- |xi| = xi+ + xi- But I'm still confused as to how to solve this...
 P: 76 Linear programming with absolute value objective function Alright, here is my graphical analysis: optimal value = 0, occurs at (0,0) and (3,2) is this correct?
 P: 76 This is what I have after doing some research online: Let X = 2x1 - 3x2 ----> Minimize |X| subject to X ≤ X' -X ≤ X' ----> Minimize X' subject to 2x1 - 3x2 ≤ X' -2x1 + 3x2 ≤ X' Question, what is X'?
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P: 5,186
 Quote by csc2iffy This is what I have after doing some research online: Let X = 2x1 - 3x2 ----> Minimize |X| subject to X ≤ X' -X ≤ X' ----> Minimize X' subject to 2x1 - 3x2 ≤ X' -2x1 + 3x2 ≤ X' Question, what is X'?
I don't understand your question. The optimal X' will be the minimal value of |2x1 - 3x2|; after all, it is >= both (2x1 - 3x2) and -(2x1 - 3x2), and you minimize it.

RGV
 P: 76 But I'm trying to solve it using IOR and I can't have a function with absolute values I thought?
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P: 5,186
 Quote by csc2iffy But I'm trying to solve it using IOR and I can't have a function with absolute values I thought?
You don't have any absolute values any more---that was the whole point of introducing the new variable X'.

Watch me compute |-3| without using absolute values:
min z
s.t.
z >= 3
z >= -3

Solution: z = 3.

RGV

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