Linear programming formulations

Keywords Call center operations Workforce management Decomposition technique Erlang C formulation Linear programming... 

Otherwise, the iterations consisting of formulation of linear approximations and the solution of the linear programming problem are continually repeated (48). 

We are now in a position to formulate the problem of minimizing the cost of MSAs. By adopting the linear-programming formulation (P6.1), one can write the following optimization program ... 

In terms of the LINGO input, the problem can be formulated via the following linear program ... 

Now, the objective function of utility cost can be minimized subject to the set of constraints (9.10)-(9.18), This formulation is a linear program which can be solved using commercially available software (e.g., LINGO). 

The above equations coupled with Eqs. (9.10)-<9.18) represent the constraints of the CHARMEN-synthesis formulation. The objective is to minimize the cost of MSAs and heating/cooling utilities. This is a linear-programming formulation whose solution determines the optimal flowrate and temperature of each substream and heating/cooling utilities. In order to demonstrate this formulation, let us consider the following example. 

The last formulation 3rields an analytical method for treating optimal flow. There are special types of linear programming problems (e.g.,... 

Equations (113) and (109)—(112) constitute the objective function and constraints of a linear programming problem. Notice that in this formulation the minimization is carried out with respect to both H(0) and Linearization is effected at the expense of increasing the number of independent (decision) variables to 1 +, vf. However, it can be shown that each... 

Glover, F., 1975. Improved linear programming formulations of nonlinear integer problems. Manag. Sci., 22(4) 455-460. 

Finally, the performance of both MILP and MINLP algorithms is strongly dependent on the problem formulations (3-110) and (3-112). In particular, the efficiency of the approach is impacted by the lower bounds produced by the relaxation of the binary variables and subsequent solution of the linear program in the branch and bound tree. A number of approaches have been proposed to improve the quality of the lower bounds, including these ... 

If the objective function and constraints in an optimization problem are nicely behaved, optimization presents no great difficulty. In particular, if the objective function and constraints are all linear, a powerful method known as linear programming can be used to solve the optimization problem (refer to Chapter 7). For this specific type of problem it is known that a unique solution exists if any solution exists. However, most optimization problems in their natural formulation are not linear. 

Formulate a linear programming model for determining how much of each product the firm should produce to maximize profit. 

Formulate the preceding problem as a linear programming problem. How many variables are there How many inequality constraints How many equality constraints How many bounds on the variables ... 

This problem is best formulated by scaling the production variables xx and x2 to be in thousands of pounds per day, and the objective function to have values in thousands of dollars per day. This step ensures that all variables have values between 0 and 10 and often leads to both faster solutions and more readable reports. We formulate this problem as the following mixed-integer linear programming problem ... 

Demand for power is 2500 megawatts (MW) in period 1 and 3500 MW in period 2. Formulate and solve this problem as a mixed-integer linear program. Define the binary variables carefully. 

The problem is to allocate optimally the crudes between the two processes, subject to the supply and demand constraints, so that profits per week are maximized. The objective function and all constraints are linear, yielding a linear programming problem (LP). To set up the LP you must (1) formulate the objective function and (2) formulate the constraints for the refinery operation. You can see from Figure El6.1 that nine variables are involved, namely, the flow rates of each of the crude oils and the four products. 

The problem involves nine optimization variables (jcc, c — 1 to 5 Qp, p = 1 to 4) in the preceding formulation. All are continuous variables. The objective function is a linear function of these variables, and so are Equations (a) and (b), hence the problem is a linear programming problem and has a globally optimal solution. 

Because the preceding formulation involves binary (Xf ) as well as continuous variables (Ci k) and has no nonlinear functions, it is a mixed-integer linear programming (MILP) problem and can be solved using the GAMS MIP solver. 

The minimization of the quadratic performance index in Equation (16.2), subject to the constraints in Equations (16.5-16.7) and the step response model such as Equation (16.1), forms a standard quadratic programming (QP) problem, described in Chapter 8. If the quadratic terms in Equation (16.2) are replaced by linear terms, a linear programming program (LP) problem results that can also be solved using standard methods. The MPC formulation for SISO control problems described earlier can easily be extended to MIMO problems and to other types of models and objective functions (Lee et al., 1994). Tuning the controller is carried out by adjusting the following parameters ... 

Ku, H. M. and I. A. Karimi. Scheduling in Serial Multiproduct Batch Processes with Finite Interstate Storage A Mixed Integer Linear Program Formulation. Ind Eng Chem Res 27 10, 1840 (1988). 

Mix bridging, 15 545 Mixed-alkali effect (MAE), 12 586-587 Mixed bauxites, 2 347 Mixed-bed columns, 14 405, 407 in ion exchange, 14 404 Mixed-bed resins, 14 412 Mixed chalcogenides, 12 359 Mixed formulation fertilizers, 11 123 Mixed-integer linear programming (MILP), 20 748 26 1023... 

Probably the most prominent approach to large-scale metabolic networks is constraint-based flux balance analysis. The steady-state condition Eq. (63) defines a linear equation with respect to the feasible flux distributions v°. Formulating a set of constraints and a linear objective function, the properties of the solution space P can be exploredusing standard techniques of linear programming (LP). In this case, the flux balance approach takes the form ... 

The above discussion shows the importance of petrochemical network planning in process system engineering studies. In this chapter we develop a deterministic strategic planning model of a network of petrochemical processes. The problem is formulated as a mixed-integer linear programming model with the objective of maximizing the added value of the overall petrochemical network. 

We demonstrate the implementation of the proposed stochastic model formulations on the refinery planning linear programming (LP) model explained in Chapter 2. The original single-objective LP model is first solved deterministically and is then reformulated with the addition of the stochastic dimension according to the four proposed formulations. The complete scenario representation of the prices, demands, and yields is provided in Table 6.2. 

The above formulation is an extension of the deterministic model explained in Chapter 5. We will mainly explain the stochastic part of the above formulation. The above formulation is a two-stage stochastic mixed-integer linear programming (MILP) model. Objective function (9.1) minimizes the first stage variables and the penalized second stage variables. The production over the target demand is penalized as an additional inventory cost per ton of refinery and petrochemical products. Similarly, shortfall in a certain product demand is assumed to be satisfied at the product spot market price. The recourse variables V [ +, , V e)+ and V e[ in... 

A standard formulation of the two-stage stochastic linear program is ... 

Linear programming is a mathematical procedure which permits you to optimize some quantity, while holding others within certain limits. For example, we may wish to minimize the cost of a multi-component formulation, while insisting that the combination of the several materials has certain required properties. 

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