Linear programming (LP) is a widely used technique to model production systems. It helps managers to make decisions in areas such as production and planning. LP provides an optimum solution as well as shadow prices for constrained resources. In addition to the optimal or most profitable operating point, it is possible to obtain profitable operating range by performing a “post-optimality analysis”. An important feature of the post optimality analysis is parametric linear programming, which investigates the effect of predetermined continuous variations in the objective function coefficients and right-hand side constraint on the optimum solution. Parametric variations provide tiers of shadow price values corresponding to every set of optimal basic feasible solutions, which are called tiered prices.

Intermediate tiered prices are widely used for advanced process control and economic optimization in petroleum refineries. Advanced Process Control (APC) targets and MV set-points are directly determined from the prices used in the objective function of the LP. The tiered price of refinery end products like aviation gasoline or motor fuel is established by the direct market value associated with it. Intermediate products like naphtha or heavy oils do not have a direct market value and their prices are determined by variations associated with operating cost, capacity limitation of equipment and, crude type. Furthermore the tiered prices for intermediate products are generated from material balance equality constraints in the LP model. Consequently, constructing tiered price values for these intermediate products becomes difficult. In this work a refinery LP model with 33-decision variables and 39-constraints obtained from literature is used to demonstrate the generation of tiered intermediate product prices for use in APC LP.

Mathematically, the LP problem is degenerate at the interface of tiers. Dual price values obtained in the optimal simplex tableau are equal to the shadow price values only when the problem is non-degenerate. Under conditions of degeneracy, dual price values are not same as shadow price values. Accurate interpretation of tiered prices for intermediate products becomes essential to make quality business decision. A two dimensional degenerate and non-degenerate LP is graphically illustrated in conjunction with simplex tableaux to improve the understanding of tiered prices.