Some problems illustrating the principles of duality

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Some problems illustrating the principles of
duality
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In this lecture we look at some problems that
uses the results from Duality theory (as
discussed in Chapter 7).
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Problem 7. Problem Set 4.2D Page 130
Consider the LPP Maximize
subject to
The following optimal tableau corresponds to
specific values of b1 and b2
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Basic z x1 x2 x3 x4
x5 Sol
1 0 0
0 1 0
a b c
7 2 -8
d 1 -1
e 0 1
150 30 10
z x1 x5

• Determine the following
• The RHS values b1, b2
• The entries a, b, c, d, e
• The optimal solution of the dual.

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(a) The optimal solution of the primal is B-1b,
where
Thus
Hence b1 30, b2 40
B-1b
(b)
Hence b 5, c -10
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a
23
5
d
e 0 as x5 is a basic variable.
(c) The optimal solution of the dual is
Note The optimal solution of the dual is also
given by d, e 5, 0.
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Problem 1(a) Problem Set 4.2E Page 131 Estimate
a range for the optimal objective value of the
following LPP Minimize
subject to
An obvious feasible solution is
x13, x20 with z 15. Thus the optimal value
? 15.
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The dual of the given LPP is the LPP Maximize
subject to
An obvious feasible solution of the dual is
y13, y21 with w 14. Thus the optimal value of
the dual ? 14.
Hence by the weak duality theorem 14 ? z? 15,
z being the optimal value of z.
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Problem 2(a) Problem set 4.2E Page 132 In
Problem 1(a), let y1, y2 be the dual variables.
Determine whether the following pairs of
primal-dual solutions are optimal
Though z 17 w, neither pair satisfy the
constraints of the respective problem. Hence not
even solutions.
Aliter Since z w ? 15, and since the given
pairs give z 17 w, they cannot be optimal.
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We shall now do some problems from section
4.2.3. The theme is How to find the optimal
solution of the dual from the primal? As we have
seen in chapter 7, the dual solution is given by
, the so-called Simplex multiplier.
In words, we can say Optimal values of dual
variables Row vector of original objective
coefficients of optimal primal variables Optimal
primal inverse.
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Now we note that in optimal tableau, the
coefficient of xj in z-Row is zj - cj
LHS of the jth dual constraint RHS of the jth
dual constraint.
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We can find the dual solution using this
equation. In your book this method of finding the
dual solution is referred to as Method 2 while
the earlier one is referred to as Method 1.
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Problem 4 Problem Set4.2C Page 126
Consider the LPP Maximize
subject to

• Write the associated dual problem.
• Given the optimal basic variables are x1 and
  x3, determine the associated optimal dual
  solution.

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• The dual problem is
• Minimize

subject to
(b)
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Hence the optimal dual solution (by method 1) is
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Problem 3 Problem Set4.2C Page 126
Consider the LPP Maximize
subject to
The primal objective row is given by
Use this information to find the associated
optimal dual solution.
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We first write the associated dual problem. Dual
Minimize
subject to
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The primal objective row is given by
Hence by method 2, the dual optimal solution
satisfy the equations
coefficient of x1 in optimal row
coefficient of x2 in optimal row
coefficient of x3 in optimal row
coefficient of x4 in optimal row
From the last two equations we get
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Problem 2 Problem Set4.2C Page 125
Consider the LPP Maximize
subject to
The primal objective row is given by
where artificial x4 and slack x5 are the starting
basic variables.
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Write the associated dual problem and determine
its optimal solution from the optimal z-equation.
Since clearly Big-M method has been used to solve
the primal, the primal (in the modified form)
is Maximize
subject to
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Hence the dual problem is Minimize
subject to
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The primal objective row is given by
Hence by method 2, the dual optimal solution
satisfy the equations
coefficient of x4 in optimal row
coefficient of x1 in optimal row
Hence we get