Inventory balance constraints

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],... 

It is very often the case that certain material types are not available in sufficient quantities to satisfy anticipated demand. To address this material scarcity it is common to consider a variety of material substitution strategies. The following material inventory balance constraint (11.9) can be used to replace constraint (11.4) to take into account material substitution. 

Equation 2.27 represents the objective function, which is the sum of the production cost, production changeover cost, and inventory shortage costs. Equation 2.28 is the demand/inventory balance constraint. If the unrestricted variable yi is replaced by the difference of two non-negative variables as... 

Equation 2.33 is the same demand/inventory balance constraint that we developed for the LP model in Section 2.15 (see Equation 2.24). 

Inventory balance constraints. The third set of constraints balances inventory at the end of each period. Net demand for Period t is obtained as the sum of the current demand Dt and the previous backlog S, i. This demand is either filled from current prodnction (in-honse... 

Column M contains workforce constraints (Eqnation 8.2), column N contains capacity constraints (Equation 8.3), column O contains inventory balance constraints (Equation 8.4), and column P contains overtime constraints (Equation 8.5). These constraints are applied to each of the six periods. 

The (apparently simplest) method using the accumulation differences (11.1.8) as measured variables in the balance constraint (11.2.4) fails in practice. It has been shown also theoretically that even small systematic errors in the measured (integrated) mass flowrates cumulate and in the end, can lead to absurd values of the holdups (states of the inventories) computed (estimated) by (11.2.9). 

Multi-period problems are usually solved in a rolling format whereby the model is solved at the begiiming of each period. For instance, at the end of the first period, the model is moved forward and the second period becomes the first period and so on. Interaction among the periods is obtained by using an inventory decision variable. For example, a buyer buys Pf units in a period t and let I, and D, be the inventory left at the end of period t and the demand in period t, respectively, then the material balance constraint for period t is... 

Equation 9.6, the material balancing constraint, states that all the requirements for a period must be satisfied (i.e., shortage and backordering are not allowed). It also specifies that the number of units purchased in any period plus any inventory carried from previous periods must be equal to demand for that period plus the inventory carried forward. Equation 9.7 ensures that the maximum the buyer can buy should not exceed the total remaining demand for that product over the planning horizon. Equation 9.8 models the capacity limit of each supplier and stipulates that a fixed cost is incurred whenever an order is placed with a supplier. The total order placed with a supplier should be less than or equal to the available capacity in that period. Note that the binary variable y is used to activate the constraint for a supplier) only if supplier) is chosen in that period. Equation 9.9 is used to model the bundling constraint, where Zj is a binary variable, one for each supplier, such that... 

P-graph, a graph-theoretic method that has proved to be efficient for process-net-work synthesis, has been utilized for designing SCs in the work of Fan et al. (2009). This approach has as advantage that the computation time does not exponentially increase with the network complexity however, further improvements may be done to better deal with inventory and mass balance constraints. 

The constraint of the pressure drop across the downcomer is graphically illustrated in Fig. 10.8. For a given solids inventory in the downcomer and given gas and solids flow rates, the pressures at the bottom of the riser and the downcomer can be determined at steady state. Under normal operating conditions (point A in the figure), the pressure drop across the riser is balanced by the pressure drop across the recirculation loop. If a small reduction in gas velocity takes place, the flow in the riser responds by moving upward along the pressure drop curve of the riser to point B. On point B of line AB, the decrease in the gas velocity causes the pressure drop across the riser to rise by SPt, which has to be balanced by the... 

Once we have fixed a flow in each recycle loop, we then determine what valve should be used to control each inventory variable. This is the material balance step in the Buckley procedure. Inventories include all liquid levels (except for surge volume in certain liquid recycle streams) and gas pressures. An inventory variable should typically be controlled with the manipulated variable that has the largest effect on it within that unit (Richardson rule). Because we have fixed a flow in each recycle loop, our choice of available valves has been reduced for inventory control in some units. Sometimes this actually eliminates the obvious choice for inventory control for that unit. This constraint forces us to look outside the immediate vicinity of the holdup we are considering. 

The dynamic simulation model has been adapted to meet the constraints of a large scale problem and of the equation solving mode of Aspen Dynamics. The final model contained more than 6000 equations. Since the change in material balance (inventory) takes place at long time scales, some substantial simplifications of the local control of units can be considered. Finally, the plantwide control problem is reduced to analyse a 3x7 system, where three outputs (concentration of impurities li, I2, and I3) should be controlled with three among five inputs (D2, SS2, Q2, D4, and Q4), in the presence of two disturbances (Fdce, X ). Because of decentralised control, at most three SISO controllers should be physically implemented. 

Base layer control can be complex, based on the specific process requirements, with both feed-forward and feedback loops, sequence control, advanced inventory control, balancing control and various types of constraint and override control. The scope of the regulatory control of the Pearl GTL plant is large as approximately 9000 manipulated variables , i.e. control valves, motors, fin-fans, etc. are used to maintain stable operation at target. With regulatory control only (i.e. without other elements in the control and automation hierarchy) the plant should be able to operate smoothly and reject disturbances, although not necessarily in the most optimal fashion. 

Also simple is the method where the (precisely unknown) state (/n ) of inventory n at time ty, i preceding time is replaced by its estimate (/ (.,), and only the measured value s (4) is reconciled at time t. in the daily balancing. Disregarding the error in (4, ) (and the corresponding probabilistic considerations), the constraint equation (11.2.11) at time ty. can be written in the form... 

Based on the steady-state optimization for each module, only one active control constraint is identified for the HDA process - the reactor inlet temperature. The nine-step procedure to generate a plantwide control structure developed by Luyben et al.[7] is now applied to each module. These steps are (i) establish the control objectives, (ii) determine the control degrees of freedom, (iii) establish energy management, (iv) set the production rate, (v) control the product quality, (vi) fix a flow in every recycle loop and control inventories, (vii) check component balances and (viii) control individual imit operations, and (ix) optimize the economics or improve the dynamic controllability. The number of control degrees of freedom identified for each module (referred to by their respective dominant unit operation) are as follows reactor 10, product column 10, and recycle column 5. 

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