.
If the primal is a minimization problem, its dual will be a maximization problem.
Columns of the constraint coefficients in the primal problem become columns of the constraint coefficients in the dual.
For an unrestricted primal variable, the associated dual constraint is an equation.
If a constraint in a maximization type of primal problem is a ‘less-than-or-equal-to’ type, the corresponding dual variable is non-negative.
The dual of the dual is primal.
An equation in a constraint of a primal problem implies the associated variable in the dual problem to be unrestricted in sign.
If a primal variable is non-negative, the corresponding dual constraint is an equation.
The objective function coefficients in the primal problem become right-hand side of constraints of the dual.
In order that dual to an LPP may be written, it is necessary that it has at least as many constraints as the number of variables.
The dual represents an alternate formulation of LPP with decision variables being implicit values.
The optimal values of the dual variables can be obtained by inspecting the optimal tableau of the primal problem as well.
Sensitivity analysis is carried out having reference to the optimal tableau alone.
All scare resources have marginal profitability equal to zero.
Shadow prices are also known as imputed values of the resources.
A constraint 3x1 – 7x2 + 13x3 – 4x4 ≥ -10 can be equivalently written as -3x1 + 7x2 – 13x3 + 4x4 ≤ 10.
If all constraints of a minimization problem are ‘≥’ type, then all dual variables are non-negative.
I and II
II and III
I and III
I, II and III
If the optimal solution to an LPP exists then the objective function values for the primal and the dual shall both be equal.
The optimal values of the dual variables are obtained from ∆j values from slack/surplus variables, in the optimal solution tableau.
An n-variable m-constraint primal problem has an m-variable n-constraint dual.
If a constraint in the primal problem has a negative bi value, its dual cannot be written.
The primal and dual have equal number of variables.
The shadow price indicates the change in the value of the objective function, per unit increase in the value of the RHS.
The shadow price of a non-binding constraint is always equal to zero.
The information about shadow price of a constraint is important since it may be possible to purchase or, otherwise, acquire additional units of the concerned resource.
Shadow prices of resources in the primal are optimal values of the dual variables.
The optimal values of the objective functions of primal and dual are the same.
If the primal problem has unbounded solution, the dual problem would have infeasibility.
All of the above.
Allocate resources optimally.
Minimize cost of operations.
Spell out relation between primal and dual.
Determine how optimal solution to LPP changes in response to problem inputs.
Nonlinear constraints
Bottlenecks in the objective function
Homogeneity
Uncertainty
Competing objectives
5 D + 7 E =< 5,000
9 D + 3 E => 4,000
5 D + 7 E = 4,000
5 D + 9 E =< 5,000
9 D + 3 E =< 5,000
1 X + 5 Y =< 750
2 X + 6 Y => 750
2 X + 5 Y = 3,000
1 X + 3 Y =< 3,000
2 X + 6 Y =>3,000
1 R + 1 W =< 8
1 R + 1 W => 30
8 R + 12 W => 30
1 R => 12
20 x (R + W) =>30
15 F + 10 C => 110
1 F + 1 C => 80
13 F + 67 C => 110
1 F => 13
13 F + 67 C =< (80/200)
A maximization function
A nonlinear maximization function
A quadratic maximization function
An uncertain quantity
A divisible additive function
Less than 5
Less than 72
Less than 512
Less than 1,024
Unlimited
Less than 5
Less than 72
Less than 512
Less than 1,024
Unlimited
Equal to 0
Less than 0
More than 0
Equal to 500
4
7
6
3
5
-1.4
0
1.4
-3.8
3.8
X - 2y ≥ -8
Y ≥ 0
-x + y ≤ 10
X + y ≤ 20
X ≥ 0
7
5
4
8
6
8
0
No solution
-8
-36
No solution
Unique solution at (2, 0)
Unique solution at (0, 0)
Infinitely many solutions
Unique solution at (0, 12)
Infinitely many solutions
Unique solution (0, 12)
No solution
Unique solution at (2, 0)
Unique solution at (0, 2)