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Preprocessing algorithm and tightening constraints for multiperiod blend scheduling: cost minimization

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Abstract

While a range of models have been proposed for the multiperiod blend scheduling problem (MBSP), solving even medium-size MBSP instances remains challenging due to the presence of bilinear terms and binary variables. To address this challenge, we develop solution methods for MBSP focusing on the cost minimization objective. We develop a novel preprocessing algorithm to calculate lower bounds on stream flows. We define product dedicated flow variables to address product specific features involved in MBSP. Bounds on stream flows and new product dedicated flow variables are then used to generate tightening constraints which significantly improve the solution time of the mixed integer nonlinear programming models as well as models based on linear approximations.

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Abbreviations

\( j \in {\mathbf{J}} \) :

Blenders

\( p \in {\mathbf{P}} \) :

Products

\( q \in {\mathbf{Q}} \) :

Properties

\( s \in {\mathbf{S}} \) :

Streams

\( t \in {\mathbf{T}} \) :

Time points/periods

\( {\mathbf{S}}_{p,q}^{\text{U}} \) :

Streams that satisfy the upper bound on property q for product p

\( {\mathbf{S}}_{p,q}^{\text{L}} \) :

Streams that satisfy the lower bound on property q for product p

\( {\mathbf{Q}}_{s,p}^{\text{S}} \) :

Properties for which stream s is the only stream that satisfies the specification for product p

\( {\mathbf{Q}}_{p}^{\text{M}} \) :

Properties for which multiple (but not all) streams satisfy the specification for product p

\( {\mathbf{Q}}^{\text{L}} \) :

Properties that have lower bounding specification

\( {\mathbf{Q}}^{\text{U}} \) :

Properties that have upper bounding specification

\( \gamma_{s}^{\text{S}} \) :

Inventory capacity for stream s

\( \gamma_{j}^{\text{J}} \) :

Inventory capacity for blender j

\( \gamma_{p}^{\text{P}} \) :

Inventory capacity for product p

\( \delta_{p,t} \) :

Amount of product p due at time point t

\( \xi_{s,t} \) :

Supply for stream s at time point t

\( \sigma_{s,q} \) :

Value of property q for stream s

\( \sigma_{p,q}^{\text{U}} \) :

Upper bounding specification on property q for product p

\( \sigma_{p,q}^{\text{L}} \) :

Lower bounding specification on property q for product p

\( \hat{\sigma }_{p,q}^{\text{U}} \) :

Value of property q that violates the upper bound for product p by the least margin

\( \hat{\sigma }_{p,q}^{\text{L}} \) :

Value of property q that violates the lower bound for product p by the least margin

\( \theta_{p,q} \) :

Value of property q of product p

\( \omega_{p} \) :

Cumulative demand for product p

\( \hat{\omega }_{s,p} \) :

Demand for stream s derived from product p

\( \bar{\omega }_{s,p,q} \) :

Demand for stream s derived from property q of product p

\( \bar{\omega }_{s,p,q}^{'} \) :

Updated demand for stream s derived from property q of product p

\( C_{q,j,t} \) :

Value of property q of the inventory in blender j during time period t

\( \tilde{F}_{s,j,t} \) :

Flow of stream s fed into blender \( j \) at time point t

\( F_{j,j',t} \) :

Flow from blender \( j \) to blender \( j' \) at time point t

\( \bar{F}_{j,p,t} \) :

Flow from blender \( j \) to product p at time point t

\( F_{j,j',t}^{S} \) :

Flow of stream s from blender \( j \) to blender \( j' \) at time point t

\( \bar{F}_{s,j,p,t}^{S} \) :

Flow of stream s from blender \( j \) to product p at time point t

\( \hat{F}_{s,p} \) :

Flow of stream s dedicated to product \( p \)

\( I_{j,t} \) :

Inventory in blender j during time period t

\( I_{s,j,t}^{S} \) :

Inventory of stream s in blender j during time period t

\( \hat{I}_{s,t} \) :

Inventory of stream s during time period t

\( \bar{I}_{p,t} \) :

Inventory of product p during time period t

\( R_{{j,j^{\prime},t}}^{\text{J}} \) :

Split fraction between flow from blender j to blender j’ at time point t and inventory of blender j at time period \( t \)

\( R_{j,p,t}^{\text{P}} \) :

Split fraction between flow from blender j to product p at time point t and inventory of blender j at time period \( t \)

\( \tilde{X}_{s,j,t} \) :

= 1 when stream s is fed into blender j at time point t

\( X_{j,j',t} \) :

= 1 when blender j feeds blender j’ at time point t

\( \bar{X}_{j,p,t} \) :

= 1 when blender j is sends product p at time point t

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  1. Department of Chemical and Biological Engineering, University of Wisconsin-Madison, 1415 Engineering Dr., Madison, WI, 53706, USA

    Yifu Chen & Christos T. Maravelias

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  1. Yifu Chen
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Chen, Y., Maravelias, C.T. Preprocessing algorithm and tightening constraints for multiperiod blend scheduling: cost minimization. J Glob Optim 77, 603–625 (2020). https://doi.org/10.1007/s10898-020-00882-3

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