An Efficient Unit-Specific Event-Based Continuous-Time MILP Formulation for Short-Term Scheduling of Multistage and Multiproduct Batch Plants
Introduction
Multistage, multiproduct batch plants with parallel units in one or more stages are commonly used in many industries such as pharmaceutical, and iron and steel industries to produce high-value, low-volume products such as specialty chemicals. Scheduling in such plants is an essential, important, and routine activity [1]. However, scheduling of such plants is extremely complicated because of its tremendous combinatorial complexity for task (order or batch) assignment, sequencing and timings.
Scheduling of batch processes especially multipurpose batch plants has received considerable amount of attention during the last three decades. Floudas and Lin [2–3], and Mendez et al. [4] presented extensive reviews on comparison of these models. Verderame et al. [5] presented a review of recent advances in planning and scheduling under uncertainty. However, scheduling of multistage and multiproduct batch plants has received limited attention in spite of their industrial significance. Castro and Grossmann [6] developed a new multiple time-grid continuous-time mixed formulation for short-term scheduling of such plants. Liu and Karimi [1] developed several novel slot-based 4-index and 3-index continuous-time formulation for scheduling a set of batches in a multistage batch plant. Although their best models perform faster than Castro and Grossmann [6], they are limited to the problem without intermediate release and due dates. Recently, Castro and Novais [7] improved the model of Castro and Grossmann [6] and proposed a multiple time-grid continuous-time MILP formulation.
In this paper, we develop an efficient unit-specific event-based continuous-time formulation for scheduling problem of multistage and multiproduct batch plants involving single product batch per stage. The computational results show that the proposed formulation is much tighter and superior to that of Castro and Novais [7] and is comparable to the formulation of Liu and Karimi [1].
Section snippets
Problem Statement
Figure 1 shows a schematic of a general multistage and multiproduct batch process. It involves S batch stages (s = 1, 2, …, S), and a total of J batch units (j = 1, 2, 3, …, J). Each stage s has Js identical or nonidentical parallel batch units. In other words, Js = {j | unit j that belongs to stage s}. It can be concluded that J = J1 + J2 + …+ JS.
Let I denote the number of orders (tasks or batches) (i = 1, 2, …, I) that the plant must process. Each unit can process Ij orders (tasks or
Mathematical Formulation
The unit-specific event-based continuous-time representation proposed by Floudas and coworkers [2–3, [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]] is utilized to develop our novel formulation for this problem since the advantages of using unit-specific event-based approaches are well established in the literature. For each unit j, we divide the scheduling horizon [0, H] into N (n = 1, 2, …, N) event points. Let Ts(j,n) and Tf(j,n) [Tf(j,n) > Ts(j,n)] denote
Comparative Studies
We solve several examples to evaluate the proposed MILP formulation using GAMS 22.6/CPLEX 11.0.0 on Dell OPTIPLEX 960 of Intel ® Xeon™ CPU 3.0 GHz with 2 GB RAM running Linux. To guarantee global optimality and do a fair comparison, we use three numbers of event points, namely n = N where the optimal solution is obtained, n = N+1, and n = I for all j in each example. The results are presented in Tables 1–2.
Conclusions
In this paper, we developed an efficient unit-specific event-based continuous-time formulation for scheduling problem of multistage and multiproduct batch plants and proposed new tighter lower bounds for some variables and several tightening constraints to improve the model performance. The computational results show that the proposed formulation is much tighter and superior to that of Castro and Novais [7] and is comparable to the formulation of Liu and Karimi [1]. Future work is to extend
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