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Linear Programming
Published in Albert G. Holzman, Mathematical Programming, 2020
The goals are expressed as constraints which are to be achieved "as close as possible." This means that the slack variable, which is added to each goal expressed as a constraint, is a free variable, and can thus take on a positive, negative, or zero value. Thus the slack variable represents the deviation from the stated goal. The objective then is to minimize the value of the slack variable associated with each goal constraint. Goal programming requires that an ordinal ranking (preemptive priority) of the objectives be made by the decision maker. If there are multiple goals for a priority level, then differential weights must be assigned to each of the goals. A modified Simplex method of linear programming attempts to satisfy the highest priority goal first, then the next highest priority goal, etc.
Optimization
Published in Slobodan P. Simonović, Managing Water Resources, 2012
From the basic interpretation of the simplex algorithm we know that the coefficient of a slack variable in Row 0 of an optimal solution represents the incremental value of another unit of the resource associated with that variable. In the preceding section we stated that the optimal value of a dual variable is the very same coefficient. Putting the two statements together, we have the following interpretation of the dual variables: The optimal value of a dual variable indicates how much the objective function changes with a unit change in the associated right-hand-side constant, provided the current optimal basis remains feasible.
More Linear Programming Models
Published in Timothy R. Anderson, Optimization Modeling Using R, 2023
The last step is where all of the inequalities are replaced by strict equality relations. The conversion of inequalities to equalities warrants a little further explanation. This is done by introducing a new, non-negative “slack” variable for each inequality. If the inequality is a ≤, then the slack variable can be thought of as filling the left-hand side to make it equal to the right-hand side so the slack variable is added to the left-hand side. If the inequality is ≥, then the slack variable is the amount that must be absorbed from the left-hand side to make it equal the right-hand side.
A new strategy for rear-end collision avoidance via autonomous steering and differential braking in highway driving
Published in Vehicle System Dynamics, 2020
Qingjia Cui, Rongjun Ding, Xiaojian Wu, Bing Zhou
The goal of the controller is to ensure the path-tracking accuracy and track the direction of the lane using the location of the centreline of the adjacent lane. Therefore, the performance of the controller is enhanced by the constraints on the lateral position, yaw angle, and force input. Based on the linear time-varying vehicle model (32), the optimal algorithm to calculate the required front tyre force is expressed as: subject to In cost function given by Equation (33a), the first term represents the performance on target following, the second term is the input force (), and the third term is the rate at which the input force () can reduce the sudden control increment. Alternatively, Q is the weighting matrix of appropriate for the output variable error. and are the weighted cost of appropriate for the control signal. To ensure that the vehicle is within the road boundaries, the constraint in Equation (33f) is applied to the output variables of the vehicle system. Additionally, slack variable ϵ is used to ensure that the optimal algorithm is feasible.
Multiobjective optimization of transit bus fleets with alternative fuel options: The case of Joinville, Brazil
Published in International Journal of Sustainable Transportation, 2020
Machado William Emiliano, Lino Costa, Sameiro Maria Carvalho, José Telhada, Edgar A. Lanzer
In this formulation, a slack variable has to be minimized (Equation (5)). The decision variables are the number of buses of type for a driving condition. The first constraint (Equation (6)) ensures that the total number of buses for each driving condition is exactly, that is, the fleet size is exactly buses per line. The weighted deviation constraints with respect to each objective are expressed by Equations (7–9). Equation (10) imposes that are non-negative and integer. Equation (11) constrains the slack variable to non-negative values.
Improved optimal design of concrete gravity dams founded on anisotropic soils utilizing simulation-optimization model and hybrid genetic algorithm
Published in ISH Journal of Hydraulic Engineering, 2021
Muqdad Al-Juboori, Bithin Datta
Concisely, to understand the process of IPA, the formulation of the optimization problem must be transferred from the general (primal) form to the standard form (dual), as shown in Table 3. Each inequality constraint, i.e. g (x), is converted to an equality constraint by adding a slack variable (si). Also, a new inequality constraint () is assumed to ensure that the slack variable is not less than zero to satisfy the original inequality constraint (Parkinson et al. 2013). The new and original equality constraints are converted to one set of equality constraints (c (x) = 0).