MILP Model

the SC network design and planning is formulated as an ( L + l)-stage stochastic MILP, where L denotes the number of events that unfold throughout the planning horizon. The structure of the SC taken as reference to develop the mathematical model is illustrated in Fig. 7.4. The model is the stochastic extension of that presented in Chap. 2. The reader is referred to the aforementioned chapter for the detailed problem formulation, but for completeness and understanding, a brief description is provided here. 

The information about the scenario tree is introduced into the model by adding two indexes to the decision variables I and hi. As an example, variable associated with a decision that is made when solely the combination of events hi is known, however the decision will be materialized at period t when event I unveils. A subscript hi below a variable also indicates that is a (/ -I- l)-stage variable. For instance, a (/ -I-1)-stage variable such as would be related to a recourse decision 

In order to follow the structure of the scenario tree, when an equation requires variables associated with previous periods (i.e., / - 1), these variables must be related to a combination of events that is ancestor of the combination of events being evaluated by the equation (/ e L j, hi e AHi fi,) so as to guarantee the non-anticipativity principle. This fact will be recalled in most of the equations presented below. 

7 Capturing Dynamics in Integrated Supply Chain Planning 

Material balances must be satisfied at each node belonging to the SC network. Equation (7.1) represents the raw material balance for each manufacturing site s in every time period t and every combination of events hi. In this equation is just taken into account the stock of previous period related to the combina- 

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MILP and MINLP models have been developed to take into account the PIS operational philosophy for assessing the efficacy of this philosophy and design, respectively. The MILP model is used to determine the effectiveness of the PIS operational philosophy by, firstly, solving the model with zero intermediate storage with and without the use of latent storage. In the illustrative example a 50% increase in the throughput was achieved. 

Constraints (11.18), (11.19), (11.20) and (11.21) constitute the linearized version of constraints (11.3). The advantage of this linearization technique is that it is exact, which implies that global optimality is assured. The disadvantage, however, is that it requires the introduction of new variables and additional constraints. Consequently, the size of the model is increased. A similar type of linearization is also necessary for constraints (11.4) in order to have an overall MILP model which can be solved exactly to yield a globally optimal solution. 

The formulation of the engineered nonlinear short-term model presented is a variant of an MINLP model described in the dissertation by Schulz [5], In this subsection, all necessary indices, parameters and variables are introduced, and the constraints and the objective function are derived. In the following section, the nonlinear formulation is linearized yielding a MILP model. In order to keep track of the variables used in the MINLP and in the MILP formulation, they are displayed in Figure 7.3 along with some key parameters. 

In this section, the numerical solutions of the MINLP-model and of the MILP-model as presented in Sections 7.4 and 7.5 are compared with respect to their solution quality (measured by the objective values) and the required solution effort (measured by the computing time). In order to compare the MILP-solution with the MINLP-solution, the optimized values for the start times of polymerizations tn, the recipe assignments W, and the total holdups Mnr are inserted into the MINLP-model and the objective is calculated. To guarantee comparability of the results, the models were stated with identical initial conditions, namely t° = 0, = 2 Vk, pf = 0 Vs, and ra = 0.4 Vs (i.e., the variables defined at the beginning of the corresponding time axes are fixed to the indicated values). For the algorithmic solution procedure, all variables were initialized by 1 (i.e., the search for optimal values starts at values of 1 ), and none of the solvers was specifically customized. 

Pinto, J.M. and Grossmann, I.E. (1996) An alternate MILP model for short-term scheduling of batch plants with preordering constraints, bid. Eng. Chem. 

A MILP model of the aggregated scheduling problem of the EPS process was proposed by Sand and Engell [16]. The model is formulated as a discrete time multi-period model where each period i e 1,..., 1 corresponds to two days. The degrees of freedom of the aggregated problem are the following discrete production decisions ... 

Table 9.3 Deterministic equivalent MILP model dimensions. ...

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],... 

A portfolio manager has 100,000 to invest in a list of 20 stocks. She estimates the return from stock i over the next year as r(i), so that if x(i) dollars are invested in stock i at the start of the year, the end of year value is [1 + r(/)] jt(/). Write an MILP model that determines the amounts to invest in each stock in order to maximize end-of-year portfolio value under the following investment policy no more than 20,000 can be invested in any stock, and if a stock is purchased at all, at least 5000 worth must be purchased. 

Gunther HO, Grunow M, Neuhaus U (2006) Realizing block planning concepts in make-and-pack production using MILP modeling and SAP APO . International Journal of Production Research 44 (18-19) 3711-3726... 

The problem is formulated as an MILP model where binary variables are used for designing the process integration network between the refineries and deciding on the production unit expansion alternatives. Linearity in the model was achieved by defining component flows instead of individual flows and associated fractions. The planning problem formulation is as follows. 

Liu, M.L. and Sahinidis, N.V. (1995) Computational trends and effects of approximations in an MILP model for process planning. Industrial ej Engineering Chemistry Research, 34, 1662. 

Sahinidis, N.V. and Grossmann, I.E. (1991a) Reformulation of multiperiod MILP models for planning and scheduling of chemical processes. Computers ei Chemical Engineering, 15, 255. 

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). 

Despite the complexity analysis results for the combinatorial nature of MILP models of the form (1), several major algorithmic approaches have been proposed and applied successfully to medium and large size application problems. In the sequel, we will briefly outline the proposed approaches and subsequently concentrate on one of them, namely, the branch and bound approach. 

The linear programming LP relaxation of the MILP model is the most frequently used type of relaxation in branch and bound algorithms. In the root node of a binary tree, the LP relaxation of the MILP model of (1) takes the form ... 

Note that Z(rcs)° >s a lower bound on the optimal solution of the MILP model of (1). Also note that if the solution of (RCS)° turns out to have all the y-variables at 0 - 1 values, then we can terminate since the LP relaxation satisfies the integrality conditions. 

The form of this MILP model for the dual subproblem is... 

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