The integrated algorithmic approach described above is based on a set of mathematical formulations that are characterized below.
6.5.1 Step 1 – Determination of the Campaign Schedules (CS model)
The campaign schedules are obtained using the non-periodic RTN discrete-time formulation (6.1) to (6.8).
By assuming that the storable intermediaries are replenished until the end of the schedule (time T) through constraints (6.4), the resulting schedule can be repeated successively and startup and shut-down phases can be avoided. The production resources include processing units, raw materials, intermediaries and final products (
), where is the campaign task . In this way, we can define campaign tasks having different products that share the set of processing units . The availability of the production resources is given by the resources balance constraints (6.1). are continuous variables that denote the availability of the resource at time interval , while are binary variables that are equal to one if task starts at time interval . The amount of resource (processing units) allocated or released by each task is specified by the parameter , which can take values during the processing time of the task . Similarly, materials are consumed and produced at the proportion of the task batch size that is modeled through the continuous variables . The resources maximum availability is guaranteed by constraints (6.2). Task batch size variables are activated through the binary variables in constraints (6.3), which also ensure that the task batch sizes are within the capacity limits of the processing units. The set of constraints (6.4) ensures the intermediaries balance at the end of the schedule. These constraints are essential to guarantee the replacement of the storable intermediaries at the end of the schedule. The net production of the final products over time is given by constraints (6.5), where is the net production of final product . And constraints (6.6) define the production bounds for , by imposing minimum and maximum amounts and , respectively. The variables domain is defined in (6.7).
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The formulation presented above assumes that the storable intermediaries are given so as to avoid startup and shutdown phases of the campaigns. Alternatively, the storable intermediaries can be determined by the optimization model, assuming that these intermediaries are not raw materials and final products, and that they have an initial equal to 0 then this intermediary is not eligible to be a storable intermediary.
(6.7.1)
(6.7.2)
∑
(6.7.3)
To have a limit on the number of storable intermediaries a new binary variable can be used. So, is equal to 1 if intermediary has been selected as storable intermediary. Constraints (6.7.3) ensure that no more than intermediaries can be storable.
Objective Function
The objective function is the maximization of the production rate and is given by expression (6.8). Several alternative schedules can be derived by solving the same model for different values of T, with T between Tmin and Tmax. Tmin is equal to the maximum processing time required to produce the stable intermediaries or the final product. Tmax is defined as the maximum acceptable duration for the campaign schedule. The selected schedule will give the maximum production for each product . Moreover, in order to calculate the minimum amount of product that can be delivered by each campaign cycle, the model was solved fixing the binary variables determined previously and assuming a minimization version of the objection function (6.8). The minimum and maximum values of represent the production bounds of product , and are used in step 2 as the minimum and maximum lot size ( ) of product at campaign task .
∑
(6.8)
6.5.2 Step 2 – Creation of the Campaign Tasks
Campaign tasks now are created taking as a basis the time chosen in Step 1.
These tasks will consume/allocate and produce/release resources according to the resources/tasks assignment made in Step 1. This approach allows modeling campaigns, as they are viewed as single production tasks, taking advantage of the uniform representation of the RTN formulation. Figure 6.5 depicts the campaign tasks for regular products PA and PB. The lot size is between the maximum and minimum allowable production taking into account the requirements of the recipes. The
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consumption and production proportions of the materials in the campaign tasks are calculated through the ratio amount of material required /amount of final product.
Therefore, the campaign task of PA has a processing time of 152 hours which results in a net production of 235 kg at the maximum lot size and PB campaign task has a processing time of 40 hours and delivers 120 kg. In step 3, it is used a RTN non-periodic formulation for scheduling all products. In order to account for lot-size-dependent processing times and also alternative units, a piecewise approximation can be done by creating multiple instances of the campaign tasks. The new campaign tasks will have different lot-size intervals that correspond to different processing times and/or units.
Figure 6.5 – Campaign tasks for the regular products PA and PB.
6.5.3 Step 3 – Detailed Scheduling Model (DS model)
Finally, a single schedule with campaign and short-term products is built by using constraints (6.9) to (6.18) and objective function (6.19). Again, we use as basis the RTN formulation.
In order to model sequence-dependent changeovers, the product index is considered in the resource availability continuous variable. Thus, variables give the processing unit availability for product at time interval . The changeover tasks are defined by binary variables that are equal to 1 if a changeover task occur on the processing unit between products and at time interval . The assignment/sequencing variables and the batch size variables are similar to the CS model. The superscript DS in the variables and sets indicate that they refer to the detailed model.
In this way, the resources balance constraints (6.9) determine the availability of the processing units for each product and time interval. The unit availability is equal to the availability in the previous time interval plus the availability resulting from
the unit’s allocation/release to/from the production or changeover tasks at time interval t.
The production tasks coefficients define the unit e allocation/release done by task k at time relative to the start of the task, and the changeovers coefficients give the allocation/release of unit from product to being the product held by the unit at time relative to the start of the changeover task. Constraints (6.10) do the initial assignment of processing units to products. Since constraints (6.9) ensure that no processing units are eliminated or created, no resource bounds on these variables are required.
Constraints (6.11) are needed to determine the materials availability . The set material resources includes raw materials, intermediaries and products, , of both campaign and short-term products. The coefficient defines the proportion of materials consumed and produced of the batch size . The continuous variables express the deliveries of the products at each the time interval t. We assume that will always have non positive values, thus no material receipts are expected to occur during the scheduling horizon . Constraints (6.12) define the minimum and maximum materials availability allowed for each time interval. Constraints (6.13) ensure that the batch size is between the minimum and maximum allowed capacities of the processing units and are just defined for the non-regular products, while constraints (6.14) define the minimum and maximum lot size of the campaign tasks (regular products).
On the demand side, the variables must be equal to zero for all materials, except for final products, see constraints (6.15), and at the time intervals different of the delivery dates , see constraints (6.16). The minimum and maximum amount of each delivery is specified by constraints (6.17). Production requirements were modeled as “soft constraints” to avoid schedule infeasibilities. The missing deliveries are expressed by the continuous variables , which are penalized in the objective function through coefficient . Practice demonstrates that this is often the case when dealing with medium and long term scheduling. Finally, expressions (6.18) express the variables domain.
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Objective Function
The objective function is given by expression (6.19) and maximizes the profit, taking into account the value of the products, inventory costs of the materials and changeovers costs. The last term introduces a penalty cost for missing deliveries.
[ ∑ ∑ branch-and-bound (B&B). However, with the increase of the number of resources or the number of tasks or, mainly, with the increase of the time periods (resulting from decreasing the duration of the time intervals or increasing the scheduling horizon), the model would lead to large optimization problems that would hardly be solved by exact methods in acceptable amount of time. Alternatively, decomposition approaches can be applied to obtain satisfactory solutions quickly.
In this work, we have decided to apply a rolling horizon approach based on the works by Dimitriadis et al. (1997) and Erdirik‐Dogan and Grossmann (2007), and the reformulation and branching strategy proposed by Velez and Maravelias (2013). The rolling horizon approach considers the detailed scheduling model (DS) and an aggregate planning model (AP), and is applied as depicted in Figure 6.6. The algorithm will progressively increase the horizon of the DS model and shrink the horizon of the AP model. The reformulation and branching strategy goal is to improve the performance of the B&B by reducing the symmetry of the scheduling solutions. Several modifications were performed in both methods so as to improve their performance.