Mixed-integer linear programming MILP

In order to make the problem solvable, a linearized process model has been derived. This enables the use of standard Mixed Integer Linear Programming (MILP) techniques, for which robust solvers are commercially available. In order to ensure the validity of the linearization approach, the process model was verified with a significant amount of real data, collected from production databases and production (shift) reports. 

Since the program (DEP) represents a mixed-integer linear program (MILP), it can be solved by commercially available state-of-the-art MILP solvers like CPLEX [3] or XPRESS-MP [4], These solvers are based on implementations of modem branch-and-bound search algorithms with cuts and heuristics. 

The SNP optimizer is based on (mixed-integer) linear programming (MILP) techniques. For a general introduction into MILP we refer to [11], An SAP APO user has no access to the mathematical MILP model. Instead, the modeling is done in notions of master data of example products, recipes, resources and transportation lanes. Each master data object corresponds to a set of constraints in the mathematical model used in the optimizer. For example, the definition of a location-product in combination with the bucket definition is translated into inventory balance constraints for describing the development of the stock level over time. Additional location-product properties have further influence on the mathematical model, e.g., whether there is a maximum stock-level for a product or whether it has a finite shelf-life. For further information on the master data expressiveness of SAP SCM we refer to [9],... 

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. 

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... 

In this chapter, we tackle the integration design and coordination of a multisite refinery network. The main feature of the chapter is the development of a simultaneous analysis strategy for process network integration through a mixed-integer linear program (MILP). The performance of the proposed model in this chapter is tested on several industrial-scale examples to illustrate the economic potential and trade-offs involved in the optimization of the network. 

This chapter explains the general representation of a petrochemical planning model which selects the optimal network from the overall petrochemical superstructure. The system is modeled as a mixed-integer linear programming (MILP) problem and illustrated via a numerical example. 

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 large number of optimization models have continuous and integer variables which appear linearly, and hence separably, in the objective function and constraints. These mathematical models are denoted as Mixed-Integer Linear Programming MILP problems. In many applications of MILP models the integer variables are 0 - 1 variables (i.e., binary variables), and in this chapter we will focus on this sub-class of MILP problems. 

In this section, we will present the formulation of Mixed-Integer Linear Programming MILP problems, discuss the complexity issues, and provide a brief overview of the solution methodologies proposed for MILP models. 

In this chapter we will discuss briefly the basics of the mixed-integer linear programming MILP model with 0-1 variables. For exposition to integer linear programming ILP with respect to all approaches the reader is referred to the excellent books of Nemhauser and Wolsey (1988), Parker and Rardin (1988), and Schrijver (1986). 

The major difficulty that arises in mixed-integer linear programming MILP problems for the form (1) is due to the combinatorial nature of the domain ofy variables. Any choice of 0 or 1 for the elements of the vector y results in a LP problem on the jc variables which can be solved for its best solution. 

In this chapter we have briefly introduced the basic notions of a branch and bound algorithmic framework, described a general branch and bound algorithm and a linear relaxation based branch and bound approach, and illustrated these ideas with a simple example. This material is intended only as a basic introduction to mixed-integer linear programming MILP problems. These MILP problems are employed as subproblems in the MINLP approaches that are discussed extensively in Chapter 6. The reader who is interested in detailed theoretical, algorithmic and computational exposition of MILP problems is directed to the excellent books of Nemhauser and Wolsey (1988), Parker and Rardin (1988), and Schrijver (1986). 

Remark 5 Note that the support functions are linear in x, and as a result v(y) will be a mixed integer linear programming MILP problem. 

Remark 7 Note that the master problem (6.20) is a mixed-integer linear programming MILP problem since it has linear objective and constraints, continuous variables (x,pOA) and 0-1 variables (y). Hence, it can be solved with standard branch and bound algorithms. 

Remark 4 The relaxed master problem is a mixed integer linear programming MILP problem which can be solved for its global solution with standard branch and bound codes. Note also that if f(x),h(x),g(x) are linear in, then we have an MILP problem. As a result, since the relaxed master is also an MILP problem, the OA/ER should terminate in 2 iterations. 

The master problem in OA and it variants OA/ER, OA/ER/AP involves linearizations of the nonlinear objective function, the vector of transformed nonlinear equalities and the original nonlinear inequalities around the optimum solution xk of the primal problem at each iteration. As a result, a large number of constraints are added at each iteration to the master problem. Therefore, if convergence has not been reached in a few iterations the effort of solving the master problem, which is a mixed-integer linear programming MILP problem, increases. 

Target (ii) was addressed rigorously by Cerda and Westerberg (1983) as a Mixed Integer Linear Programming MILP transportation model and by Papoulias and Grossmann (1983) as an MILP transshipment model. Both models determine the minimum number of matches given the minimum utility cost. 

The appropriate definition of the lower and upper bounds can have a profound effect on the computational effort of solving the model P2. In fact, the tighter the bounds, the less effort is required, even though the same solution can be obtained for arbitrarily large Uij. Finally, the nonnegativity and the top and bottom residual constraints are also linear. The variables are a mixed set of continuous and binary variables. Therefore, P2 corresponds to a mixed-integer linear programming MILP transshipment model. 

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