Abstract
Water is considered a significant resource in process industries. It is essential for planners to target and optimize the use of water as an external resource for industrial operations. Such optimization problems account for uncertainties related to internal resources and must be handled to provide solutions for real plants of industrial relevance. In this paper, these parametric uncertainties are addressed, while targeting resources for continuous and flexible schedule batch process. The proposed robust counterpart formulations include resource minimization constraints for continuous and batch processes to satisfy the demand. Three different robust optimization methodologies are adapted and extended to handle parametric uncertainties associated with internal resources. Assuming bounded and known uncertainty, the resultant formulations are then implemented to literature examples, and the results are compared with the deterministic formulation. The results show that the formulation proposed by Bertsimas and Sim is the most appropriate model for the defined problem because it preserves the linearity and provides a mechanism to control the degree of conservatism, guaranteeing feasibility. The proposed formulations are also explained using illustrative examples estimating the additional 4.04% and 11% requirement of resource in continuous and batch process, respectively, to handle uncertainty with a known risk. This model will assist the planner to decide the resource requirement under uncertain conditions and do the necessary preparation accordingly, and thus, it immunes the process against uncertainties to satisfy demands.
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Abbreviations
- \(N_{\rm{sr}}\) :
-
Number of sources
- \(N_{\rm{sk}}\) :
-
Number of sinks
- \(s \in S\) :
-
Any state
- \(s_{\rm{sr}} \in S_{\rm{sr}}\) :
-
Source state \((S_{\rm{sr}} \subset S)\)
- \(s_{\rm{sk}} \in S_{\rm{sk}}\) :
-
Sink state \((S_{\rm{sk}} \subset S)\)
- \(i \in I\) :
-
Unit
- \(k \in K\) :
-
Event point
- R :
-
External resource requirement for continuous process
- \(f\left( {s_{\rm{sr}} ,s_{\rm{sk}} } \right)\) :
-
Flow supplied from source ‘\(s_{\rm{sr}}\)’ to demand related to sink ‘\(s_{\rm{sk}}\)’
- \(f_{w} \left( {s_{\rm{sr}} } \right)\) :
-
Flow supplied from source ‘\(s_{\rm{sr}}\)’ to waste
- \(f_{r} \left( {s_{\rm{sk}} } \right)\) :
-
Flow supplied from external resource R to demand related to sink ‘\(s_{\rm{sr}}\)’
- \(u_{c} \left( {s_{\rm{sr}} ,s_{\rm{sk}} } \right),\) \(v_{c} \left( {s_{\rm{sr}} ,s_{\rm{sk}} } \right)\) :
-
Additional variables associated with RO 2 for continuous process formulation
- \({\text{z}}_{f}^{c} ,q_{f}^{c} \left( {s_{\rm{sk}} } \right),\) \({\text{z}}_{c}^{c} ,q_{c}^{c} \left( {s_{\rm{sr}} ,s_{\rm{sk}} } \right)\) :
-
Additional variables associated with RO 3 for continuous process formulation
- \(F_{\rm{sr}} \left( {s_{\rm{sr}} } \right)\) :
-
Flow available from source related to ‘\(s_{\rm{sr}}\)’
- \(\overline{{F_{\rm{sr}} }} \left( {s_{\rm{sr}} } \right)\) :
-
Nominal value of flow available from source related to ‘\(s_{\rm{sr}}\)’
- \(\widehat{{F_{\rm{sr}} }}\left( {s_{\rm{sr}} } \right)\) :
-
Variation amplitude from \(\overline{{F_{\rm{sr}} }} \left( {s_{\rm{sr}} } \right)\)
- \(c_{\rm{sr}} \left( {s_{\rm{sr}} } \right)\) :
-
Contaminant concentration of source related to ‘\(s_{\rm{sr}}\)’
- \(\overline{{c_{\rm{sr}} }} \left( {s_{\rm{sr}} } \right)\) :
-
Nominal value of contaminant concentration of source related to ‘\(s_{\rm{sr}}\)’
- \(\widehat{{c_{\rm{sr}} }}\left( {s_{\rm{sr}} } \right)\) :
-
Variation amplitude from \(\overline{{c_{\rm{sr}} }} \left( {s_{\rm{sr}} } \right)\)
- \(F_{\rm{sk}} \left( {s_{\rm{sk}} } \right)\) :
-
Flow demand related to sink ‘\(s_{\rm{sk}}\)’
- \(c_{\rm{sk}} \left( {s_{\rm{sk}} } \right)\) :
-
Concentration demand related to sink ‘\(s_{\rm{sk}}\)’
- \(f\left( {s_{\rm{sr}} ,s_{\rm{sk}} } \right)\) :
-
Flow supplied from source ‘\(s_{\rm{sr}}\)’ to sink ‘\(s_{\rm{sk}}\)’
- \(f_{r} \left( {s_{\rm{sk}} } \right)\) :
-
Flow supplied from external resource ‘R’ to sink ‘\(s_{\rm{sk}}\)’
- \(\Omega_{f}^{c} ,\Omega_{c}^{c}\) :
-
Additional parameters associated with RO 2 for continuous process formulation
- \(\Gamma_{f}^{c} ,\Gamma_{c}^{c}\) :
-
Budget parameter associated with RO 3 for continuous process formulation
- MM:
-
Any large number
- \(\tau_{\rm{sr}} \left( {s_{\rm{sr}} ,i} \right)\) :
-
Duration of source related to state ‘\(s_{\rm{sr}}\)’ and unit ‘\(i\)’
- \(\tau_{\rm{sk}} \left( {s_{\rm{sk}} ,i} \right)\) :
-
Duration of sink’s demand related to state ‘\(s_{\rm{sk}}\)’ in unit ‘\(i\)’
- \(F_{\rm{sr}} \left( {s_{\rm{sr}} ,i,k} \right)\) :
-
Flow available from source related to state ‘\(s_{\rm{sr}}\)’ and unit ‘\(i\)’
- \(\overline{{F_{\rm{sr}} }} \left( {s_{\rm{sr}} ,i,k} \right)\) :
-
Nominal value of flow available from source related to state ‘\(s_{\rm{sr}}\)’ and unit ‘\(i\)’, for deterministic case \(F_{\rm{sr}} \left( {s_{\rm{sr}} ,i,k} \right) = \overline{{F_{\rm{sr}} }} \left( {s_{\rm{sr}} ,i,k} \right)\)
- \(\widehat{{F_{\rm{sr}} }}\left( {s_{\rm{sr}} ,i,k} \right)\) :
-
Variation amplitude from \(\overline{{F_{\rm{sr}} }} \left( {s_{\rm{sr}} ,i,k} \right)\)
- \(F_{\rm{sk}} \left( {s_{\rm{sk}} ,i^{\prime},k^{\prime}} \right)\) :
-
Flow demand of sink related to state ‘\(s_{\rm{sk}}\)’ in unit ‘\(i\)’
- \(c_{\rm{sr}} \left( {s_{\rm{sr}} ,i} \right)\) :
-
Contaminant concentration available from source related to state ‘\(s_{\rm{sr}}\)’ in unit ‘\(i\)’
- \(\overline{{c_{\rm{sr}} }} \left( {s_{\rm{sr}} ,i} \right)\) :
-
Nominal value of contaminant concentration available from source related to state ‘\(s_{\rm{sr}}\)’ in unit ‘\(i\)’, for deterministic case \(c_{\rm{sr}} \left( {s_{\rm{sr}} ,i} \right) = \overline{{c_{\rm{sr}} }} \left( {s_{\rm{sr}} ,i} \right)\)
- \(\widehat{{c_{\rm{sr}} }}\left( {s_{\rm{sr}} ,i} \right)\) :
-
Variation amplitude from \(\overline{{c_{\rm{sr}} }} \left( {s_{\rm{sr}} ,i} \right)\)
- \(c_{\rm{sk}} \left( {s_{\rm{sk}} ,i} \right)\) :
-
Maximum contaminant concentration limit accepted by sink related to state ‘\(s_{\rm{sk}}\)’ in unit ‘\(i\)’
- \(\Omega_{f}^{b} ,\Omega_{c}^{b}\) :
-
Additional parameters associated with RO 2 for batch process formulation
- \(\Gamma_{f}^{b} ,\Gamma_{c}^{b}\) :
-
Budget parameter associated with RO 3 for batch process formulation
- \(R\left( {s_{\rm{sk}} ,i^{\prime},k^{\prime}} \right)\) :
-
Resource requirement for demand related to \(\left( {s_{\rm{sk}} ,i^{\prime},k^{\prime}} \right)\)
- \(T_{p} \left( {s,i,k} \right)\) :
-
Time at which state ‘s’ appears in unit ‘\(i\)’ at event point ‘\(k\)’
- \(T_{\rm{sr},s} \left( {s_{\rm{sr}} ,i,k} \right)\) :
-
Time at which source related to state ‘\(s_{\rm{sr}}\)’ starts in unit ‘i’ at event point ‘k’.
- \(T_{\rm{sr},e} \left( {s_{\rm{sr}} ,i,k} \right)\) :
-
Time at which source related to state ‘\(s_{\rm{sr}}\)’ ends in unit ‘i’ at event point ‘k’.
- \(T_{\rm{sk},s} \left( {s_{\rm{sk}} ,i,k} \right)\) :
-
Time at which sink’s demand related to state ‘\(s_{\rm{sk}}\)’ starts in unit ‘i’ at event point ‘k’
- \(T_{\rm{sr},e} \left( {s_{\rm{sr}} ,i,k} \right)\) :
-
Time at which sink’s demand related to state ‘\(s_{\rm{sk}}\)’ ends in unit ‘i’ at event point ‘k’
- \(X\left( {s_{\rm{sk}} ,s_{\rm{sr}} ,i,i^{\prime},k,k^{\prime}} \right)\) :
-
Fraction of time when the source related to (\(s_{\rm{sr}} ,i,k\)) to supply the sink’s demand related to \(\left( {s_{\rm{sk}} ,i^{\prime},n^{\prime}} \right)\) to the total duration of the source, where \(i^{\prime} \in I, n^{\prime} \in N\)
- \(f_{av} \left( {s_{\rm{sk}} ,s_{\rm{sr}} ,i,i^{\prime},k,k^{\prime}} \right)\) :
-
Flow available from a source related to \((s_{\rm{sr}} ,i,k\)) to supply the sink’s demand related to \(\left( {s_{\rm{sk}} ,i^{\prime},n^{\prime}} \right)\), where \(i^{\prime} \in I, n^{\prime} \in N\)
- \(f_{sup} \left( {s_{\rm{sk}} ,s_{\rm{sr}} ,i,i^{\prime},k,k^{\prime}} \right)\) :
-
Flow supplied from a source related to (\(s_{\rm{sr}} ,i,k\)) to the sink’s demand related to \(\left( {s_{\rm{sk}} ,i^{\prime},n^{\prime}} \right)\), where \(i^{\prime} \in I, n^{\prime} \in N\)
- \(y_{w} \left( {s_{\rm{sk}} ,s_{\rm{sr}} ,i,i^{\prime},k,k^{\prime}} \right)\) :
-
Binary variable denoting availability of source related to (\(s_{\rm{sr}} ,i,k\)) related to sink’s demand related to \(\left( {s_{\rm{sk}} ,i^{\prime},n^{\prime}} \right)\), where \(i^{\prime} \in I, n^{\prime} \in N\)
- \(u_{b} \left( {s_{\rm{sk}} ,s_{\rm{sr}} ,i,i^{\prime},k,k^{\prime}} \right), v_{b} \left( {s_{\rm{sk}} ,s_{\rm{sr}} ,i,i^{\prime},k,k^{\prime}} \right)\) :
-
Additional variables associated with RO 2 for batch process formulation
- \(z_{f}^{b} \left( {s_{\rm{sr}} ,i,k} \right)\) \(z_{c}^{b} \left( {s_{\rm{sk}} ,i^{\prime},k^{\prime}} \right)\) \(q_{f}^{b} \left( {s_{\rm{sr}} ,i,k} \right)\) \(q_{c}^{b} \left( {s_{\rm{sk}} ,s_{\rm{sr}} ,i,i^{\prime},k,k^{\prime}} \right)\) :
-
Additional variables associated with RO 3 for batch process formulation
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Acknowledgements
The authors would like to thank the Department of Science and Technology-Science and Engineering Research Board, India (DST-SERB) and Indian Institute of Technology, Patna for providing the research funding for this project under Grant No. ECR/2018/000197.
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Kumawat, P.K., Chaturvedi, N.D. Robust resource targeting in continuous and batch process. Clean Techn Environ Policy 24, 273–288 (2022). https://doi.org/10.1007/s10098-021-02118-8
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DOI: https://doi.org/10.1007/s10098-021-02118-8