Feasibility Constraints

This constraints ensures that only one task can take place in a unit at a particular time point. 

Constraints (3.12) and (3.13) ensure that all the tasks take place within the time  

This constraints ensures that the maximum capacity of the intermediate storage units is not exceeded. 

The model in the above form does not take into account the possibility of using latent storage, i.e. PIS operational philosophy. There are a number of additional constraints needed to fully capture this operational philosophy. 

Constraint (3.15) states that the mass entering a process unit for latent storage must be between the minimum and maximum capacities of the unit. Furthermore, it ensures that mass can only enter the unit if the binary variable associated with latent storage is active for unit j at time point p. 

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These options may involve major cost or feasibility constraints and would require conducting an analysis prior to implementation. 

The final group of sequencing and scheduling constraints comprise of feasibility constraints and time horizon constraints. 

The final scheduling constraints are the feasibility constraints and time horizon constraints. Constraint (9.68) ensures that a processing unit can only process one task... 

Constraints (10.15) states that the amount of cold duty required by unit j at any point along the time horizon of interest is comprised of external cold utility and cold duty from heat integration with another unit j. Constraints (10.16) is similar to constraints (10.15) and applies to unit j requiring heating. Constraints (10.17) is a feasibility constraints, which ensures that in the absence of heat integration all the heat duty requirements of either unit j or j are satisfied by external utilities. The upper bound on the amount of heat exchanged between unit j and unit j will always be the minimum of the required cold and hot utilities as captured in Constraint (10.17 ). 

Constraints (11.16) is a feasibility constraints which ensures that if a unit is not integrated with storage, then the associated duty should not exist. 

The explicit feasibility constraints of (MASTER) are given by the linear first-stage constraints in (9.4.2). In a classical penalty function approach, the explicit feasibility constraints are relaxed while the violation of these constraints is considered by an additional penalty term in the fitness function. However, this method would waste valuable CPU time since the MILP subproblems (SUB) have to be solved also for the fitness evaluation of infeasible individuals. A similar method which does not require the solution of the MILP subproblems for infeasible individuals is the use of a modified objective Junction that separates the objective and the feasibility... 

This sub-problem considers the structural constraints that result in generation of feasible molecular structures. Odele Machietto structural feasibility constraints are used. The sub-problem is a function of the binary variables alone. 

The evaluation of Step 1 serves as a basis for the question "How could operations be improved " Step 2, "Description of the Solution" re-counts the development of a desired future state for the repective business. In each process, all persons directly involved find aspects needing improvement. Ideal solutions, free of feasibility constraints, will be described showing how optimised operations could run eliminating the current weak points. 

In terms of the actual HEN feasibility constraints (including the equality constraints), NLP (4) can be expressed more explicitly as (Saboo et al., 1987a)... 

The boundary of R is determined by t i = 0. Individual segments in the boundary of R are determined by/m = 0, m G M. Values of the uncertain variables 6 lying inside feasible region R allow the control variables z to be adjusted so that all the feasibility constraints can be satisfied. For values of 8 lying outside the feasible region, the control variables cannot be adjusted to satisfy all the feasibility constraints. 

If X d) > 0, then at least one of the feasibility constraints is violated somewhere in the uncertainty range. 

Using the definition of x given by Eq. (7a), a formulation in terms of the feasibility constraints is... 

A sufficient condition that the RI be determined by a vertex critical point is that the feasible region R be convex. (Of course, a special case of this is when all the feasibility constraints are linear see Section III,B.) Unfortunately, when flow rates or heat transfer coefficients are included in the uncertainty range, the feasible region can be nonconvex (see Examples 1 and 2 and Section III,C,3). Thus, current algorithms for calculating the RI are limited to temperature uncertainties only. 

The feasible region defined by constraints (a)-(c) is the convex region below the dashed line in Fig. 9. All other feasibility constraints for the HEN (for the heating only case) lie outside this region. 

A HEN is resilient in a specified uncertainty range 0 if and only if X(d) 0. If (d) > 0, then at least one of the feasibility constraints fm is violated somewhere in the uncertainty range. Geometrically, the resilience test determines whether uncertainty range 0 lies entirely within feasible region R. 

To test for operability of the HEN in the specified uncertainty range, the active constraint strategy is applied to the flexibility index at the stage of structure (without the energy recovery constraint). First, constraints (A1)-(A5) and (B1)-(B4) are developed for this network. Since there are nine equations and 12 unknowns, there exist three control variables which have been selected to be zx = w 2, z2 = w 2, z3 = T4 (see Fig. 20). Using the information from the gradients of the feasibility constraints with respect to the control variables, four active sets of constraints are identified. Then, solving an NLP for each active set of constraints, it was found... 

Here x represents a vector of n continuous variables (e.g., flows, pressures, compositions, temperatures, sizes of units), and y is a vector of integer variables (e.g., alternative solvents or materials) h(x,y) = 0 denote the to equality constraints (e.g., mass, energy balances, equilibrium relationships) g(x,y) < 0 are the p inequality constraints (e.g., specifications on purity of distillation products, environmental regulations, feasibility constraints in heat recovery systems, logical constraints) f(x,y) is the objective function (e.g., annualized total cost, profit, thermodynamic criteria). 

Remark 4 In the considered example we have a fixed EM AT = 10 since we were given that HRAT — TIAT = EM AT = 10. Note, however, that EMAT participates linearly in the feasibility constraints. As a result, we can relax it in a straightforward fashion, which is to write the feasibility constraints as... 

These can be obtained from the feasibility constraints for this example and are... 

Note, however, that cases I and II cannot take place because if T56 < 350, then the feasibility constraint... 

Note that the feasibility constraints are incorporated in the lower bound constraints of the temperatures. 

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