Data-driven feasibility analysis for the integration of planning and scheduling problems

Abstract

A framework for the integration of planning and scheduling using data-driven methodologies is proposed. First, the constraints at the planning level related to the scheduling problem are identified. This includes the feasibility of production targets assigned to each planning period (which are equivalent to scheduling horizons). Then, classification methods are used to identify feasible regions from large amounts of scheduling data, and an algebraic equation for the predictor is obtained. The predictor is incorporated in the planning problem, and the integrated problem is solved to optimality. Computational studies are presented to demonstrate the performance of the proposed framework, and results show that the approach is more efficient than current practices in the integration of planning and scheduling problems.

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Abbreviations

i ∈ I :

Processing tasks

j ∈ J :

Units

k ∈ K :

Scheduling time points/periods

s ∈ S :

States

t ∈ T :

Planning time points/periods

m :

Set of data points used to generate classifiers

I j :

Set of tasks \(i\) that can be performed in unit j

J i :

Set of units \(j\) that can performed task \(i\)

IC s :

Set of tasks \(i\) that consume state \(s\)

IP s :

Set of tasks \(i\) that produce state \(s\)

S FP :

Set of states related to finished products

S R :

Set of states related to raw materials

C s :

Storage capacity for state \(s\)

P st :

Production targets for state \(s\)

SH :

Scheduling horizon

\(V_{ij}^{min}\) :

The minimum capacity for unit \(j\) processing task \({\text{i}}\)

\(V_{ij}^{max}\) :

The maximum capacity for unit \(j\) processing task \(i\)

ρ is :

Stoichiometric coefficients related to production/consumption of state \(s\) in task \(i\)

\(\tau\) :

Scheduling discretization time

\(B_{ijk}\) :

Batch size of task \(i\) processed in unit \(j\) starting at time point \(k\)

\(X_{ijk}\) :

Binary variable which is \(1\) if task \(i\) is processed in unit \(j\) starting at time point \(k\)

\(W_{sk}\) :

Inventory level of state \(s\) at time point \(k\)

\(D_{st}\) :

Demand for product \(s\) in planning period \(t\)

\(hc_{s}\) :

Holding cost for state \(s\)

\(uc_{s}\) :

Unmet demand penalty for product \(s\)

\(rc_{s}\) :

Cost of raw material \(s\)

\(\gamma_{ss'}\) :

Stoichiometric coefficients related to production of state \(s\) and raw material \(s'\)

\(PH\) :

Planning horizon

\(Ch_{t}\) :

Holding cost at period \(t\)

\(Cp_{t}\) :

Processing cost at period \(t\)

\(Cu_{t}\) :

Unmet demand penalty at period \(t\)

\(Inv_{st}\) :

Inventory levels for product \(s\) in period \(t\)

\(P_{st}\) :

Production targets for state \(s\) in planning period \(t\)

\(U_{st}\) :

Unmet demand \(U_{st}\) for each product \(s\) and period \(t\)

\({\mathbf{x}}_{m}\) :

Vector of production targets used to train the classifiers

\({\mathbf{X}}\) :

The set of vectors \({\mathbf{x}}_{m}\)

\(y_{m}\) :

Label that classifies vector \({\mathbf{x}}_{m}\) in feasible or infeasible

\({\mathbf{y}}\) :

Label vector, \({\mathbf{y}} = \left\{ {y_{1} ,y_{2} , \ldots ,y_{M} } \right\}\) used to train the classifiers

\(d\) :

Number of products (features) in the planning problem

\(M\) :

Number of data points used to obtain classifiers

\({\mathbf{St}}\) :

Training set

\({\mathbf{Test}}\) :

Testing set

\(G\left( \cdot \right)\) :

Measure of impurity

\(j\) :

Features, or the components of vector \({\mathbf{x}}^{m}\)

\(t_{split}\) :

A threshold for partitioning the data

\(I\left( \cdot \right)\) :

Improvement gain

\(N_{p}\) :

Data at the parent node

\(N_{L}\) :

Data at the left node

\(N_{R}\) :

Data at the right node

\(p\left( {N_{m} } \right)\) :

Relative frequency of class 1 in node \(N_{m}\)

\(q\) :

Fraction of instances going left

\(\theta\) :

Candidate split

b :

Bias

\(N_{sv}\) :

Number of support vectors

v :

Set of support vectors

w :

Weight

\(\alpha\) :

Lagrangian multiplier

\(\xi\) :

Slack variables

\({\mathbf{b}}_{input}\) :

Bias in the input layer

\({\mathbf{b}}_{hidden}\) :

Bias in the output layer

\(Loss\) :

Loss function

N :

Number of nodes in the hidden layer

R :

Predictor

w input :

Weights in the input layer

\({\mathbf{w}}_{hidden}\) :

Weights in the output layer

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Acknowledgements

LSD gratefully acknowledges financial support from CNPQ—Conselho Nacional de Desenvolvimento Científico e Tecnológico—Brazil. MGI acknowledges financial support from NSF under Grant CBET 1159244, Grant CBET 1839007 and Grant CBET 1547171.

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Correspondence to Marianthi G. Ierapetritou.

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Appendices

Appendix 1: Examples data

See Figs. 13, 14, 15, Tables 7, 8, 9, 10, 11 and 12.

Fig. 13
figure13

Demand for two-dimensional problem

Fig. 14
figure14

Demand for three-dimensional problem

Fig. 15
figure15

Demand for the seven-dimensional problem

Table 7 Data for two-dimensional problem
Table 8 Data for two-dimensional problem
Table 9 Data for three-dimensional problem
Table 10 Data for three-dimensional problem
Table 11 Data for seven-dimensional problem
Table 12 Data for seven-dimensional problem

Appendix 2: Classifiers data

See Tables 13 and 14.

Table 13 Parameters for the support vector predictor in case study 1
Table 14 Parameter for the neural network predictor in case study 1

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Dias, L.S., Ierapetritou, M.G. Data-driven feasibility analysis for the integration of planning and scheduling problems. Optim Eng 20, 1029–1066 (2019). https://doi.org/10.1007/s11081-019-09459-w

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Keywords

  • Scheduling of production
  • Production planning
  • Integrated planning and scheduling
  • Feasibility analysis
  • Supervised learning