Standard Maximization Problem in linear programming

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Maximization Problem
A standard maximization problem in n unknowns is a linear programming problem in which we are required to maximize the objective function: Z = c1x1 + c2x1 + . . . + cn xn subject to constraints of the form a11x1 + a12x2 +. . .+ a1nxn ≤ b1 a21x1 + a22x2 +. . .+ a2nxn ≤ b2 . . . am1x1 + am2x2 +. . .+ amn xn ≤ bm and further restriction of the form x1 ≥ 0, x2 ≥ 0 , . . . , xn ≥ 0 Note that the inequality here must be a " ≤ " and not "=" or "≥"

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first a convert all the constraints and the objective functions in to a linear equations .
then it become x1 + x2 = 6
x1=3
x2=3
2×1+3×2 = 3
After substitution in equations to find the points
(3,3) is the point pf interaction of x1=3 and x2=3
and the corner points in the feasible region is (3,0) , (5,0) .

then substitution in the objective function z = 5×1 + 3×2 .

the maximum value at x=5 and x2=0
maximun value is 25

the minimum value at x1=3 and x2=0 .
minimum value is 15

i’am i right ??
could any one guide me .

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  • TomLiotta
    Interesting, but do you have a problem? -- Tom
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  • nolyasery
    i didn't get it very well . could any one explain it to me please
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  • nolyasery
    thanks no need now i understood what i missed . =)
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  • nolyasery
    could you just help me in these question because i want to make sure about my answer and understanding .   Consider the graphical representation of  the following LPP    Maximize   Z =   5x1  +  3 x2 Subject  to           x1 +   x2  6,                              x1   3                                       x2  3                           2x1 + 3 x2  3                            x1  0 , x2  0.   Answer the following questions: (a)  In each of the following cases indicate if the solution space ( the feasible region)  has one point, infinite number of points, or no points. 1.    The constraints are as given above. 2.    The constraint  x1 +   x2  6 is changed to  x1 +   x2  5. 3.    The constraint  x1 +   x2  6 is changed to  x1 +   x2  7. (b)  For  each  case  in (a) , determine the number of  feasible extreme points, if any. (c)   For  the  cases  in  (a) in which a feasible solution exists  , determine the  maximum values  of  Z . (d)  Answer part (c) for minimize Z.
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  • nolyasery
    here is more clear picture of the question http://www.4shared.com/photo/kYVHM_QB/Screen_shot_2012-09-29_at_8172.html
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  • TomLiotta
    That is the statement of a practice problem. In order to help, we need to see the work that you have done. We can then point to errors or clarify your understanding. But we can't do the work for you. For example, show your objective row values first. -- Tom
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  • TomLiotta
    You can try practicing at this site if you want to before working on your problem. -- Tom
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  • nolyasery
    first a convert all the constraints and the objective functions in to a linear equations . then it become x1 + x2 = 6 x1=3 x2=3 2x1+3x2 = 3 After substitution in equations to find the points (3,3) is the point pf interaction of x1=3 and x2=3 and the corner points in the feasible region is (3,0) , (5,0) . then substitution in the objective function z = 5x1 + 3x2 . the maximum value at x=5 and x2=0 maximun value is 25 the minimum value at x1=3 and x2=0 . minimum value is 15 i'am i right ?? could any one guide me .
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  • TomLiotta
    You don't show your matrices, so it's hard to tell how you got your results. But it looks like you're doing okay so far.   Let's see:   Z = 5X1 + 3X2   With constraints:   X1 + X2 ≤ 6 X1 ≥ 3 X2 ≥ 3 2X1 + 3X2 ≥ 3   Additional constraints (that are true but logically unnecessary):   X1 ≥ 0 ; X2 ≥ 0   Personally, I find the first three constraints a little odd. Both X1 and X2 can only equal 3. There are no other values that can possibly satisfy:   X1 + X2 ≤ 6   Where did you get the problem?   Tom
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