Integer Linear Programming

Ilfiire 10.14 Cascade diagram for Example 10.7, Awing temperature intervals, heat balances, and I Rsiduals, AH, and in kilowatts. 

y is a binary variable that equals unity when a match exists between hot stream i and cold stream j, and is zero otherwise. The objective is to minimize the number of matches, and hence, the objective function sums over all of the possible matches, with the weighting coefficient, Wy, increased as certain matches become less desirable. Constraints (MEP.l) and (MILP.2) are the energy balances for each of the K temperature intervals where i, the hot 

Constraints (MILP.3) place bounds on the heat to be transferred when hot stream i and cold stream j are matched. Of course, when yij = 0, there is no match. However, when = 1, there is an upper bound on the heat that can he transferred between the two streams in all of the temperature intervals (X Qijt). This upper hound, Uy, is the minimum of the heat that can be released by hot stream / [0/ = and that which can be taken up by cold 

Constraints (MILP.4) assure that all of the residuals and rates of heat transfer are greater than or equal to zero and define the binary variables. Finally, constraints (MILP.5) indicate that the residuals at the lower and upper bounds of the temperature intervals are zero, and hence, all streams that exchange energy must have their temperatures within the bounds of the temperature intervals, including the hot and cold utilities. This is one of the principal departures from the temperature intervals in the previous analysis (see Examples 10.2 and 10.6), where the hot and cold utilities are not included in the temperature intervals. 

The next example is provided to illustrate the creation and solution of a typical MILP for MER design. 

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In this way the problem is reformulated as an MILP (Mixed Integer Linear Programming) problem. Readers who are interested in the problem of discrete sizing are referred to the paper of Voudoris and Grossmann (1992). 

When a linear programming problem is extended to include integer (binary) variables, it becomes a mixed integer linear programming problem (MILP). Correspondingly,... 

One of the approaches that can be used in design is to carry out structural and parameter optimization of a superstructure. The structural optimization required can be carried out using mixed integer linear programming in the case of a linear problem or mixed integer nonlinear programming in the case of a nonlinear problem. Stochastic optimization can also be very effective for structural optimization problems. 

Mixed Integer Linear Programming If the objective and constraint functions are all linear, then (3-84) becomes a mixed integer linear programming problem given by... 

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. 

A mixed-integer linear programming approximation can be derived following a problem specific approach. 

Both the mixing process and the approximation of the product profiles establish nonconvex nonlinearities. The inclusion of these nonlinearities in the model leads to a relatively precise determination of the product profiles but do not affect the feasibility of the production schedules. A linear representation of both equations will decrease the precision of the objective but it will also eliminate the nonlinearities yielding a mixed-integer linear programming model which is expected to be less expensive to solve. 

A continuous-time mixed integer linear programming model for... 

I. E. (1997) A mixed-integer linear programming model for shortterm scheduling of single-stage multiproduct batch plants with parallel lines. Ind. Eng. 

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 hybrid algorithm is in general suitable for any two-stage stochastic mixed-integer linear program with integer requirements in the first-stage and in the... 

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],... 

EXAMPLE 9.1 BRANCH-AND-BOUND ANALYSIS OF AN INTEGER LINEAR PROGRAM... 

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 ... 

The NLP solver used by GAMS in this example is CONOPT2, which implements a sparsity—exploiting GRG algorithm (see Section 8.7). The mixed-integer linear programming solver is IBM s Optimization Software Library (OSL). See Chapter 7 for a list of commercially available MILP solvers. 

Generalized Benders decomposition (GBD), derived in Geoffrion (1972), is an algorithm that operates in a similar way to outer approximation and can be applied to MINLP problems. Like OA, when GBD is applied to models of the form (9.2)-(9.5), each major iteration is composed of the solution of two subproblems. At major iteration K one of these subproblems is NLP(y ), given in Equations (9.6)-(9.7). This is an NLP in the continuous variables x, with y fixed at y The other GBD subproblem is an integer linear program, as in OA, but it only involves the... 

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. 

Raman, R. and I. E. Grossmann. Symbolic Integration of Logic in Mixed Integer Linear Programming Techniques for Process Synthesis. Comput Chem Eng 17 909-928, (1993). 

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. 

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... 

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