As in the model solved initially, the following is the LP model
Maximize Z = $42.13*(x_{11} + x_{12} + x_{13} + x_{14}) + $38.47*(x_{21} + x_{22} + x_{23} + x_{24}) + $27.87*(x_{31} + x_{32} + x_{33} + x_{34})
With subject to constraints
Production Constraints
x_{11} + x_{21} + x_{31} = 3814
x_{12} + x_{22} + x_{32} = 2666
x_{13} + x_{23} + x_{33} = 4016
x_{14} + x_{24} + x_{34} = 1300
PN quality constraints
107x_{11} + 93x_{12} + 87x_{13} + 108x_{14 }≥ 100*(x_{11} + x_{12} + x_{13} + x_{14})
107x_{21} + 93x_{22} + 87x_{23} + 108x_{24 }≥ 91*(x_{21} + x_{22} + x_{23} + x_{24})
RVP quality constraints
5x_{11} + 8x_{12} + 4x_{13} + 21x_{14 }≤ 7*(x_{11} + x_{12} + x_{13} + x_{14})
5x_{21} + 8x_{22} + 4x_{23} + 21x_{24 }≤ 7*(x_{21} + x_{22} + x_{23} + x_{24})
Non-negativity constraint
x_{ij }≥ 0
where x_{ij }is the amount of i, i = 1 (Alkylate), 2 (catalytic-cracked), 3 (straight-run), 4 (isopentane) to be mixed in j, j = 1 (Blend 1 or Avgas A), 2 (Blend 2 or Avgas B), 3 (Raw)
Dual problem
Associated with every LP problem is a related dual problem. If the objective in the original problem is maximization, then the objective in the dual is the minimization of a related (but different) function. Conversely, an original minimization problem has a related dual maximization problem. Thus in our case, since the original problem is a maximization, the dual is a minimization of a related function.
Associated with each constraint of the original problem is a dual variable. Since the original problem had eight constraints, the dual problem will have eight variables, namely w_{1}, w_{2}, w_{3}, w_{4}, w_{5}, w_{6}, w_{7} and w_{8}. It is not required to have a variable for non-negativity constraint of the original problem. As we see the last four constraints (quality constraints) in the original problem can be still solved as follows
107x_{11} + 93x_{12} + 87x_{13} + 108x_{14 }≥ 100x_{11} + 100x_{12} + 100x_{13} + 100x_{14}
7x_{11} - 7x_{12} - 13x_{13} + 8x_{14 }≥ 0
Similarly,
107x_{21} + 93x_{22} + 87x_{23} + 108x_{24 }≥ 91x_{21} + 91x_{22} + 91x_{23} + 91x_{24}
16x_{21} + 2x_{22} - 4x_{23} + 17x_{24 }≥ 0
5x_{11} + 8x_{12} + 4x_{13} + 21x_{14 }≤ 7x_{11} + 7x_{12} + 7x_{13} + 7x_{14}
-2x_{11} + 1x_{12} - 3x_{13} + 14x_{14 }≤ 0
-2x_{21} + 1x_{22} - 3x_{23} + 14x_{24 }≤ 0
With the quality constraints solved as above, the objective of the dual problem is minimization as below. It is to be noted that if the constraints in the original problem had greater than or equal to sign, they will be taken as negative values in the minimization problem. The constraints with equal to and less than or equal to signs will be taken as positive values or as such in the original problem. Since there are 12 decision variables in the original problem, the dual will have 12 constraints as below, excluding the non-negativity constraint.
Minimize Z = 3814w_{1} + 2666w_{2} + 4016w_{3}+ 1300w_{4} - 0w_{5} - 0w_{6} + 0w_{7} + 0w_{8}
With subject to constraints
w_{1 }- 7w_{5} - 2w_{7} ≥ 42.13
w_{2 }+ 7w_{5} + w_{7} ≥ 42.13
w_{3 }+ 13w_{5} - 3w_{7} ≥ 42.13
w_{4 }- 8w_{5} + 14w_{7} ≥ 42.13
w_{1 }- 16w_{6} - 2w_{8} ≥ 38.47
w_{2 }- 2w_{6} + w_{8} ≥ 38.47
w_{3 }+ 4w_{6} - 3w_{8} ≥ 38.47
w_{4 }- 17w_{6} + 14w_{8} ≥ 38.47
w_{1} ≥ 27.87
w_{2} ≥ 27.87
w_{3} ≥ 27.87
w_{4} ≥ 27.87
w_{i} ≥ 0
The dual variables w_{i}, i = 1, 2, 3, 4 represent the marginal value of production constraints, i = 5, 6 represent the marginal value of PN quality constraints and i = 7, 8 represent the marginal value of RVP quality constraints. On solving this using excel solver, we get the same objective as $481,742.9. If we look more closely into the sensitivity report of the original problem, we can notice that the values of decision variables obtained in the dual problem will be none other than the shadow prices of the constraints in the original problem.