Multimodal Processes Rescheduling: Cyclic Steady States Space Approach

Table 2

Constraints (9) determining the multimodal processes behavior of the SCCP from Figure 2.

Constraints for stream 𝑚𝑃^11	Constraints for stream 𝑚𝑃^12	Constraints for stream 𝑚𝑃^13
𝑚𝑥^11,1=max𝑚𝑥^41,2+Δ𝑡𝑚−𝑚𝛼,𝑚𝑥^11,9+𝑚𝑡^11,9−𝑚𝛼𝒳11,1; 𝑚𝑥^11,2=max𝑚𝑥^41,3+Δ𝑡𝑚−𝑚𝛼,𝑚𝑥^11,1+𝑚𝑡^11,1𝒳11,1; 𝑚𝑥^11,3=max𝑚𝑥^41,4+Δ𝑡𝑚−𝑚𝛼,𝑚𝑥^11,2+𝑚𝑡^11,2𝒳11,3; 𝑚𝑥^11,4=max𝑚𝑥^41,5+Δ𝑡𝑚−𝑚𝛼,𝑚𝑥^11,3+𝑚𝑡^11,3𝒳11,4; 𝑚𝑥^11,5=max𝑚𝑥^12,4+Δ𝑡𝑚,𝑚𝑥^11,4+𝑚𝑡^11,4𝒳11,5; 𝑚𝑥^11,6=max𝑚𝑥^41,7+Δ𝑡𝑚,𝑚𝑥^11,5+𝑚𝑡^11,5𝒳11,6; 𝑚𝑥^11,7=max𝑚𝑥^13,8+Δ𝑡𝑚,𝑚𝑥^11,6+𝑚𝑡^11,6𝒳11,7; 𝑚𝑥^11,8=max𝑚𝑥^41,9+Δ𝑡𝑚,𝑚𝑥^11,7+𝑚𝑡^11,7𝒳11,8; 𝑚𝑥^11,9=max𝑚𝑥^41,9+𝑚𝑡^41,9+Δ𝑡𝑚,𝑚𝑥^11,8+𝑚𝑡^11,8𝒳11,9	𝑚𝑥^12,1=max𝑚𝑥^42,2+Δ𝑡𝑚−𝑚𝛼,𝑚𝑥^12,9+𝑚𝑡^12,9−𝑚𝛼𝒳12,1; 𝑚𝑥^12,2=max𝑚𝑥^42,3+Δ𝑡𝑚−𝑚𝛼,𝑚𝑥^12,1+𝑚𝑡^12,1𝒳12,1; 𝑚𝑥^12,3=max𝑚𝑥^41,6+Δ𝑡𝑚,𝑚𝑥^12,2+𝑚𝑡^12,2𝒳12,3; 𝑚𝑥^12,4=max𝑚𝑥^42,5+Δ𝑡𝑚−𝑚𝛼,𝑚𝑥^12,3+𝑚𝑡^12,3𝒳12,4; 𝑚𝑥^12,5=max𝑚𝑥^13,4+Δ𝑡𝑚,𝑚𝑥^12,4+𝑚𝑡^12,4𝒳12,5; 𝑚𝑥^12,6=max𝑚𝑥^13,5+Δ𝑡𝑚,𝑚𝑥^12,5+𝑚𝑡^12,5𝒳12,6; 𝑚𝑥^12,7=max𝑚𝑥^13,6+Δ𝑡𝑚,𝑚𝑥^12,6+𝑚𝑡^12,6𝒳12,7; 𝑚𝑥^12,8=max𝑚𝑥^42,9+Δ𝑡𝑚,𝑚𝑥^12,7+𝑚𝑡^12,7𝒳12,8; 𝑚𝑥^12,9=max𝑚𝑥^42,9+𝑚𝑡^42,9+Δ𝑡𝑚,𝑚𝑥^12,8+𝑚𝑡^12,8𝒳12,9	𝑚𝑥^13,1=max𝑚𝑥^43,2+Δ𝑡𝑚−𝑚𝛼,𝑚𝑥^13,11+𝑚𝑡^13,11−𝑚𝛼𝒳13,1; 𝑚𝑥^13,2=max𝑚𝑥^43,3+Δ𝑡𝑚−𝑚𝛼,𝑚𝑥^13,1+𝑚𝑡^13,1𝒳13,1; 𝑚𝑥^13,3=max𝑚𝑥^42,6+Δ𝑡𝑚,𝑚𝑥^13,2+𝑚𝑡^13,2𝒳13,3; 𝑚𝑥^13,4=max𝑚𝑥^42,7+Δ𝑡𝑚−𝑚𝛼,𝑚𝑥^13,3+𝑚𝑡^13,3𝒳13,4; 𝑚𝑥^13,5=max𝑚𝑥^42,8+Δ𝑡𝑚,𝑚𝑥^13,4+𝑚𝑡^13,4𝒳13,5; 𝑚𝑥^13,6=max𝑚𝑥^43,7+Δ𝑡𝑚,𝑚𝑥^13,5+𝑚𝑡^13,5𝒳13,6; 𝑚𝑥^13,7=max𝑚𝑥^41,8+Δ𝑡𝑚,𝑚𝑥^13,6+𝑚𝑡^13,6𝒳13,7; 𝑚𝑥^13,8=max𝑚𝑥^43,9+Δ𝑡𝑚,𝑚𝑥^13,7+𝑚𝑡^13,7𝒳13,8; 𝑚𝑥^13,9=max𝑚𝑥^43,10+Δ𝑡𝑚,𝑚𝑥^13,8+𝑚𝑡^13,8𝒳13,9; 𝑚𝑥^13,10=max𝑚𝑥^43,11+Δ𝑡𝑚,𝑚𝑥^13,9+𝑚𝑡^13,9𝒳13,10; 𝑚𝑥^13,11=max𝑚𝑥^41,11+𝑚𝑡^41,11+Δ𝑡𝑚,𝑚𝑥^13,10+𝑚𝑡^13,10𝒳13,11
𝑚𝑥^11,1∈𝒳^11,1=𝑥^11,2(𝑙)∣𝑥^11,2(𝑙)=𝑥^11,2+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^11,2∈𝒳^11,2=𝑥^11,3(𝑙)∣𝑥^11,3(𝑙)=𝑥^11,3+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^11,3∈𝒳^11,3=𝑥^11,4(𝑙)∣𝑥^11,4(𝑙)=𝑥^11,4+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^11,4∈𝒳^11,4=𝑥^12,7(𝑙)∣𝑥^12,7(𝑙)=𝑥^12,7+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^11,5∈𝒳^11,5=𝑥^12,8(𝑙)∣𝑥^12,8(𝑙)=𝑥^12,8+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^11,6∈𝒳^11,6=𝑥^32,5(𝑙)∣𝑥^32,5(𝑙)=𝑥^32,5+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^11,7∈𝒳^11,7=𝑥^32,6(𝑙)∣𝑥^32,6(𝑙)=𝑥^32,6+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^11,8∈𝒳^11,8=𝑥^24,7(𝑙)∣𝑥^24,7(𝑙)=𝑥^24,7+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^11,9∈𝒳^11,9=𝑥^24,8(𝑙)∣𝑥^24,8(𝑙)=𝑥^24,8+𝑙⋅𝛼,𝑙∈ℤ	𝑚𝑥^12,1∈𝒳^12,1=𝑥^15,1(𝑙)∣𝑥^15,1(𝑙)=𝑥^15,1+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^12,2∈𝒳^12,2=𝑥^15,2(𝑙)∣𝑥^15,2(𝑙)=𝑥^15,2+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^12,3∈𝒳^12,3=𝑥^15,3(𝑙)∣𝑥^15,3(𝑙)=𝑥^15,3+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^12,4∈𝒳^12,4=𝑥^25,7(𝑙)∣𝑥^25,7(𝑙)=𝑥^25,7+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^12,5∈𝒳^12,5=𝑥^25,8(𝑙)∣𝑥^25,8(𝑙)=𝑥^25,8+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^12,6∈𝒳^12,6=𝑥^16,3(𝑙)∣𝑥^16,3(𝑙)=𝑥^16,3+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^12,7∈𝒳^12,7=𝑥^16,4(𝑙)∣𝑥^16,3(𝑙)=𝑥^16,4+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^12,8∈𝒳^12,8=𝑥^17,3(𝑙)∣𝑥^17,3(𝑙)=𝑥^17,3+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^12,9∈𝒳^12,9=𝑥^17,4(𝑙)∣𝑥^17,4(𝑙)=𝑥^17,4+𝑙⋅𝛼,𝑙∈ℤ	𝑚𝑥^13,1∈𝒳^13,1=𝑥^18,1(𝑙)∣𝑥^18,1(𝑙)=𝑥^18,1+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^13,2∈𝒳^13,2=𝑥^18,2(𝑙)∣𝑥^18,2(𝑙)=𝑥^18,2+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^13,3∈𝒳^13,3=𝑥^18,3(𝑙)∣𝑥^18,3(𝑙)=𝑥^18,3+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^13,4∈𝒳^13,4=𝑥^26,4(𝑙)∣𝑥^26,2(𝑙)=𝑥^26,4+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^13,5∈𝒳^13,5=𝑥^26,4(𝑙)∣𝑥^26,2(𝑙)=𝑥^26,4+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^13,6∈𝒳^13,6=𝑥^14,3(𝑙)∣𝑥^14,3(𝑙)=𝑥^14,3+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^13,7∈𝒳^13,7=𝑥^14,4(𝑙)∣𝑥^14,3(𝑙)=𝑥^14,4+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^13,8∈𝒳^13,8=𝑥^22,5(𝑙)∣𝑥^22,5(𝑙)=𝑥^22,5+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^13,9∈𝒳^13,9=𝑥^22,6(𝑙)∣𝑥^22,6(𝑙)=𝑥^22,6+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^13,10∈𝒳^13,10=𝑥^13,3(𝑙)∣𝑥^13,3(𝑙)=𝑥^13,3+𝑙⋅𝛼,𝑙∈ℤ; 𝑚𝑥^13,11∈𝒳^13,11=𝑥^13,4(𝑙)∣𝑥^13,4(𝑙)=𝑥^13,4+𝑙⋅𝛼,𝑙∈ℤ
Constraints of the rest of streams 𝑚𝑃^21, 𝑚𝑃^31, 𝑚𝑃^41, 𝑚𝑃^22, 𝑚𝑃^32, 𝑚𝑃^42, 𝑚𝑃^23, 𝑚𝑃^33, and 𝑚𝑃^43 are defined in the same way.