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Formulating and Solving Linear Programs
Published in Craig A. Tovey, Linear Optimization and Duality, 2020
When an LP is solved, both the optimal primal and dual values are found. The meaning of the primal variables had better be clear. If it isn't, you didn't define your variables properly in your formulation. The dual variables have meaning, too. Recall that there is a dual variable for each constraint in an LP. The value of the dual variable is the cost of enforcing its corresponding constraint (see Definition 1.8), that is, the effect on the objective function of changing the right-hand side of the constraint. When the constraint is an upper bound on the availability of a resource, the dual variable equals the value to you of an additional unit of the resource. That is why it is called the shadow price of the resource. It is what you would be willing to pay for an additional unit. If the market price is lower than the shadow price, it would be advantageous to purchase more of that resource. If the market price is higher, it would be advantageous (in the LP model) to sell a unit of the resource. In general, the shadow price of a constraint is the rate at which the objective function gets better as the constraint is relaxed.
Economic analysis of projects
Published in J.C. Edison, Infrastructure Development and Construction Management, 2020
In certain cases, the market price may not accurately reflect the economic value of the goods or services, for instance, when: (i) a buyer or seller has undue influence onthe price; or (ii) where there are other limitations to efficient pricing. Under these conditions, it is common to apply a shadow price. A shadow price is a non-market-determined price that has been calculated to approximate the economic value of the resources involved in the provision of goods or services.14 It is used in project appraisal when: there is strong evidence ofnon-performing markets; oradministrated prices are far from matching supply and demand.
Base Case Results
Published in S. C. Littlechild, K. G. Vaidya, Energy Strategies for the UK, 2019
S. C. Littlechild, K. G. Vaidya
Associated with each constraint in a linear programme is a ‘dual variable’ or ‘shadow price’, which indicates the change in the value of the objective function that would be caused by a small incremental change in the constant on the ‘right-hand side’ of that constraint. For example, the shadow price associated with the constraint requiring coal output to be not less than coal demand in a particular year may be interpreted as the cost of meeting the demand for one extra PJ of coal in that year - in effect, this is the marginal cost of supplying coal. This calculation of marginal cost, it should be noted, is based upon the cheapest way of meeting the additional demand, taking into account whatever changes are appropriate in production and investment patterns for other fuels also. The shadow prices on the oil, gas and electricity demand constraints may be interpreted likewise.
Approach for integrated product variant allocation and configuration adaption of global production networks featuring post-optimality analysis
Published in International Journal of Production Research, 2022
Jan Hochdörffer, Felix Klenk, Thomas Fusen, Benjamin Häfner, Gisela Lanza
Within its validity range, a shadow price shows the behaviour of the objective function value in case of extended underlying capacity constraints (Dauber, Shim, and Siegel 2013). Hence, potential capacity adjustments can be assessed with this method. In the case of a minimisation programme, negative shadow prices represent a reduction of the objective value and thus a cost saving in the case of a capacity expansion on constraint . The maximum value up to which a capacity can be increased so that the previous basic solution remains valid is equal to the upper end of the validity range. The amount by which the capacity can be increased until this value is reached thus corresponds to the difference between the upper end of the validity period and the original capacity limitation . Applying this potential for improvement to the specially relaxed programme, analogously to the slack variables, results in a change of the objective value , which is illustrated by the following equation (Hochdörffer 2018):
Convex nonparametric least squares and stochastic semi-nonparametric frontier to estimate the shadow prices of PM2.5 and NOx for Taiwan’s transportation modes
Published in International Journal of Sustainable Transportation, 2021
Huey-Kuo Chen, Yi-Hsiu Lin, Chia-Yen Lee
Shadow price, which is defined as the economic virtual price of goods, activities, and services. Since it has no market price and is difficult to estimate, it can be treated as the opportunity cost which represents the most valuable choice of those that were not taken when making a choice. In pollutant emission, the shadow price of undesirable outputs is interpreted as (1) the opportunity cost of abating one additional unit of undesirable output in terms of the loss of desirable output; (2) the marginal abatement costs (MAC) of pay for abating one additional unit of undesirable output. The latter one refers to the concept of “user fees”, i.e., the firms (the decision-making units, DMUs) which produce pollution, are responsible for the external costs incurred, because the costs of undesirable outputs can be viewed as input costs that would decrease desirable outputs. Shadow prices estimation is useful and may provide reference information when making policy analysis (Zhou et al., 2014).
Demand response versus storage flexibility in energy: multi-objective programming considerations
Published in Optimization, 2021
Wim van Ackooij, Ana Paula Chorobura, Claudia Sagastizábal, Hasnaa Zidani
In Linear Programming, shadow prices are Lagrange multipliers of certain constraint, giving the rate of change of the objective function when the constraint varies. We extend this sensitivity concept to the multi-objective setting, to determine how the variation of the first objective function affects the Pareto solution.