Shadow prices

Lagrange multipliers are often referred to as shadow prices, adjoint variables, or dual variables, depending on the context. Suppose the variables are at an optimum point for the problem. Perturb the variables such that only constraint hj changes. We can write... 

The multiplier tells how the optimal value of the objective func tion changes for this small change in the value of a constraint while holding all the other constraints at zero. It is for this reason that they are often called shadow prices. 

As the goal is to minimize the objective function, releasing the constraint into the feasible region must not decrease the objective function. Using the shadow price argument above, it is evident that the multipher must be nonnegative (Ref. 177). 

The range information is of two basic types. Shadow prices are available for items not in the solution basis, such as an equipment capacity that was completely used up with none remaining. The shadow price is the incremental value of additional capacity and ranges are provided to show the capacity range to which this value applies. 

Cell Name Final Value Shadow Price Constraint R.H. Side Allowable Increase Allowable Decrease... 

In Problem 7.1 what are the shadow prices for incremental production of gasoline, kerosene, and fuel oil Suppose the profit coefficient for crude 1 is increased by 10 percent and crude 2 by 5 percent. Which change has a larger influence on the objective function ... 

At convergence the Lagrange multipliers have some physical meaning similar to the "shadow prices" discussed in Section 1.2. The interested reader may consult (refs. 28-29) where nonideality, treatment of condensed phases, numerical difficulties and other problems are also discussed. Handbooks like (ref. 38) contain the necessary standard Gibbs free energy data for a great number of substances. 

Select the suitable criterion for the nicotine - tar data investigated in Sections 1.S.2, 1.8.3 and 3.1. Inspecting the shadow prices in the minimax estimation omit the most suspectible point and repeat the estimations by the different methods. Discuss the sensitivity of the various estimates with respect to omitting this point. 

The clearing tariff equal to c includes a shadow price, equal to c a, which is the consumers net willingness to pay for a marginal capacity expansion. 

The Lagrange multipliers in a constrained nonlinear optimization problem have a similar interpretation to the dual variables or shadow prices in linear programming. To provide such an interpretation, we will consider problem (3.3) with only equality constraints that is,... 

Direct prices do not take into account the effect a decision in one part of a plant may have on the irreversibilities in another. Marginal and shadow prices do this but are more complicated to compute. They depend upon the system of equations (and their first derivatives with respect to the variables of interest) rather than upon only the states of various zones. The mathematical description of a thermodynamic process requires the specification of a set of "equations of constraint", represented here by the set, [4>.=0]. The thermodynamic performance and stream variables are divided into two sets, state and decision variables, represented by [x.] and [y ], and each of the defining functions, [4.], is expressed in terms of these variables. If the objective function, 4, (whether it is an energy objective or a cost objective) is similarly expressed, a Lagrangian may be defined according to ... 

The solution of the set of linear equations represented by (13) determines the shadow prices. In terms of these prices the marginal prices are given by ... 

If the basis matrix can be diagonalized, the marginal and shadow prices may be found in a direct way. 

The matrices indicated in equation (15) are known as Jacobians. The properties of Jacobians have been of interest to thermodynamicists for many years and may be used to find an interesting interpretation of the meanings of the shadow prices. Using the methods described in (2,8), define ... 

We have found that shadow prices are easier to interpret if they are expressed as multiples of exergy prices by using the equation ... 

Table VI. Evaluating regeneration using shadow prices, example 2...

The Lagrangian multipliers may be interpreted as prices which reveal the economic value associated with a differential change in the corresponding state variables, and are referred to as shadow prices (5). 

System Solution. The shadow prices, Xj , are evaluated using Equation 10. Table IV lists the algebraic solution for the shadow prices (made very easy by choice of diagonalized constraint equations). The marginal prices 0 are evaluated with Equation 15. Again, the choice of fixed parameters and decision variables makes it possible to solve Equation 15 explicitly for each marginal price. Table V lists these expressions. 

Table IV. Shadow Price Equations for the Cogeneration System...

The solution procedure requires the designer to select a feasible set of decision variables y for the first iteration. Once this initial set of five decision variables has been chosen, the entire design (for that iteration) is fixed and the set of state variables x, and cost estimates are determined. With values assigned to all the state and decision variables, the set of shadow prices SP(i), is evaluated, and in turn, the set of marginal prices, PM(i), is determined. These marginal prices are then used to direct the iteration as described by Equation 16. 

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