A probabilistic approach to performance analysis of various subsystems in an Indian sugar plant

Abstract

The performance analysis of the evaporation and crystallization unit of a sugar plant is explored in the current study. It is an important unit requiring more concentration to increase its performance. The probabilistic approach is used to formulate the transition diagram of the evaporation and crystallization unit. Then, the first-order differential equations are obtained using the mnemonic rule and further solved recursively. The availability model of the concerned unit is obtained at all full and reduced capacity states. The failure and repair rate values are placed into the developed availability model to determine the decision matrices. It gives the availability levels of different subsystems of the concerned unit. The criticalities of subsystems are based on availability levels mentioned in decision matrices to determine the maintenance priorities. It is observed that the evaporator subsystem is most critical in which the availability increased from 0.7260 to 0.8888, i.e. (16.20%). In contrast, the sugar grader is the least crucial subsystem, with availability enhanced from 0.8740 to 0.8893 (i.e., 1.53%). The results show that the evaporator subsystem must be given top priority, and the sugar grader subsystem must be given the least priority from the maintenance outlook.

Appendices

Appendix

Equations 1–30

$${{\mathrm{P}}^{\mathrm{^{\prime}}}}_{\mathrm{S}0}(\dot{t)}+{\Phi }_{17}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}+{\Phi }_{18}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}+{\Phi }_{19}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}+{\Phi }_{20}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}+{\Phi }_{21}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}={\upmu }_{17}{\mathrm{P}}_{\mathrm{S}1}(\dot{t)}+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}2}(\dot{t)}+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}3}(\dot{t)}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}4}(\dot{t)}+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}5}(\dot{t)}$$

(1)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}1}(\dot{t)}+{\Phi }_{17}{\mathrm{P}}_{\mathrm{S}1}(\dot{t)}+{\Phi }_{18}{\mathrm{P}}_{\mathrm{S}1}(\dot{t)}+{\Phi }_{19}{\mathrm{P}}_{\mathrm{S}1}(\dot{t)}+{\Phi }_{20}{\mathrm{P}}_{\mathrm{S}1}(\dot{t)}+{\Phi }_{21}{\mathrm{P}}_{\mathrm{S}1}(\dot{t)}+{\upmu }_{17}{\mathrm{P}}_{\mathrm{S}1}(\dot{t)}= {\upmu }_{17}{\mathrm{P}}_{\mathrm{S}6}(\dot{t)}+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}7}(\dot{t)}+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}8}(\dot{t)}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}9}(\dot{t)} +{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}10}(\dot{t)}+{\Phi }_{17}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}$$

(2)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}2}(\dot{t)}+{\Phi }_{17}{\mathrm{P}}_{\mathrm{S}2}(\dot{t)}+{\Phi }_{18}{\mathrm{P}}_{\mathrm{S}2}(\dot{t)}+{\Phi }_{19}{\mathrm{P}}_{\mathrm{S}2}(\dot{t)}+{\Phi }_{20}{\mathrm{P}}_{\mathrm{S}2}(\dot{t)}+{\Phi }_{21}{\mathrm{P}}_{\mathrm{S}2}(\dot{t)}+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}2}(\dot{t)}={\upmu }_{17}{\mathrm{P}}_{\mathrm{S}11}(\dot{t)}+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}12}(\dot{t)}+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}13}(\dot{t)}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}14}(\dot{t)}+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}15}(\dot{t)}+{\Phi }_{18}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}$$

(3)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}3}(\dot{t)}+{\Phi }_{17}{\mathrm{P}}_{\mathrm{S}3}(\dot{t)}+{\Phi }_{18}{\mathrm{P}}_{\mathrm{S}3}(\dot{t)}+{\Phi }_{19}{\mathrm{P}}_{\mathrm{S}3}(\dot{t)}+{\Phi }_{20}{\mathrm{P}}_{\mathrm{S}3}(\dot{t)}+{\Phi }_{21}{\mathrm{P}}_{\mathrm{S}3}(\dot{t)}+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}3}(\dot{t)}={\upmu }_{17}{\mathrm{P}}_{\mathrm{S}16}(\dot{t)}+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}17}(\dot{t)}+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}18}(\dot{t)}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}19}(\dot{t)}+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}20}(\dot{t)}+{\Phi }_{19}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}$$

(4)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}4}(\dot{t)}+{\Phi }_{17}{\mathrm{P}}_{\mathrm{S}4}(\dot{t)}+{\Phi }_{18}{\mathrm{P}}_{\mathrm{S}4}(\dot{t)}+{\Phi }_{19}{\mathrm{P}}_{\mathrm{S}4}(\dot{t)}+{\Phi }_{20}{\mathrm{P}}_{\mathrm{S}4}(\dot{t)}+{\Phi }_{21}{\mathrm{P}}_{\mathrm{S}4}(\dot{t)}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}4}(\dot{t)}={\upmu }_{17}{\mathrm{P}}_{\mathrm{S}21}(\dot{t)}+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}22}(\dot{t)}+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}23}(\dot{t)}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}5}(\dot{t)}+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}24}(\dot{t)}+{\Phi }_{20}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}$$

(5)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}5}(\dot{t)}+{\Phi }_{17}{\mathrm{P}}_{\mathrm{S}5}(\dot{t)}+{\Phi }_{18}{\mathrm{P}}_{\mathrm{S}5}(\dot{t)}+{\Phi }_{19}{\mathrm{P}}_{\mathrm{S}5}(\dot{t)}+{\Phi }_{20}{\mathrm{P}}_{\mathrm{S}5}(\dot{t)}+{\Phi }_{21}{\mathrm{P}}_{\mathrm{S}5}(\dot{t)}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}5}(\dot{t)}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}28}(\dot{t)}+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}29}(\dot{t)}+{\Phi }_{20}{\mathrm{P}}_{\mathrm{S}4}(\dot{t)}$$

(6)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}6}(\dot{t)}+{\upmu }_{17}{\mathrm{P}}_{\mathrm{S}6}(\dot{t)} ={\Phi }_{17}{\mathrm{P}}_{\mathrm{S}1}(\dot{t)}$$

(7 )

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}7}(\dot{t)}+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}7}(\dot{t)}={\Phi }_{14}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}$$

(8)

$${{\mathrm{P}}^{\mathrm{^{\mathrm{^{\prime}}}}}}_{\mathrm{S}8}(\dot{t)}+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}8}(\dot{t)}={\Phi }_{19}{\mathrm{P}}_{\mathrm{S}1}(\dot{t)}$$

(9)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}9}(\dot{t)}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}9}(\dot{t)}={\Phi }_{20}{\mathrm{P}}_{\mathrm{S}1}(\dot{t)}$$

(10 )

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}10}(\dot{t)}+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}10}(\dot{t)}={\Phi }_{21}{\mathrm{P}}_{\mathrm{S}1}(\dot{t)}$$

(11 )

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}11}(\dot{t)}+{\upmu }_{17}{\mathrm{P}}_{\mathrm{S}11}(\dot{t)}={\Phi }_{17}{\mathrm{P}}_{\mathrm{S}2}(\dot{t)}$$

(12)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}12}(\dot{t)}+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}12}(\dot{t)}={\Phi }_{18}{\mathrm{P}}_{\mathrm{S}2}(\dot{t)}$$

(13)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}13}(\dot{t)}+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}13}(\dot{t)}={\Phi }_{19}{\mathrm{P}}_{\mathrm{S}2}(\dot{t)}$$

(14)

$${{\mathrm{P}}^{\mathrm{^{\prime}}}}_{\mathrm{S}14}(\dot{t)}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}14}(\dot{t)}={\Phi }_{20}{\mathrm{P}}_{\mathrm{S}2}(\dot{t)}$$

(15)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}15}(\dot{t)}+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}15}(\dot{t)}={\Phi }_{21}{\mathrm{P}}_{\mathrm{S}2}(\dot{t)}$$

(16)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}16}(\dot{t)}+{\upmu }_{17}{\mathrm{P}}_{\mathrm{S}16}(\dot{t)}={\Phi }_{17}{\mathrm{P}}_{\mathrm{S}3}(\dot{t)}$$

(17)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}17}(\dot{t)}+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}17}(\dot{t)}={\Phi }_{18}{\mathrm{P}}_{\mathrm{S}312 }(\dot{t)}$$

(18)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}18}(\dot{t)}+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}18}(\dot{t)}={\Phi }_{19}{\mathrm{P}}_{\mathrm{S}3}(\dot{t)}$$

(19)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}19}(\dot{t)}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}19}(\dot{t)}={\Phi }_{20}{\mathrm{P}}_{\mathrm{S}3}(\dot{t)}$$

(20)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}20}(\dot{t)}+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}20}(\dot{t)}={\Phi }_{21}{\mathrm{P}}_{\mathrm{S}3}(\dot{t)}$$

(21)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}21}(\dot{t)}+{\upmu }_{17}{\mathrm{P}}_{\mathrm{S}21}(\dot{t)}={\Phi }_{17}{\mathrm{P}}_{\mathrm{S}4}(\dot{t)}$$

(22)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}22}(\dot{t)}+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}22}(\dot{t)}={\Phi }_{18}{\mathrm{P}}_{\mathrm{S}4}(\dot{t)}$$

(23)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}23}(\dot{t)}+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}23}(\dot{t)}={\Phi }_{19}{\mathrm{P}}_{\mathrm{S}4}(\dot{t)}$$

(24)

$${{\mathrm{P}}^{\mathrm{^{\prime}}}}_{\mathrm{S}24}(\dot{t)}+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}24}(\dot{t)}={\Phi }_{21}{\mathrm{P}}_{\mathrm{S}4}(\dot{t)}$$

(25)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}25}(\dot{t)}+{\upmu }_{17}{\mathrm{P}}_{\mathrm{S}25}(\dot{t)}={\Phi }_{17}{\mathrm{P}}_{5\mathrm{S}}(\dot{t)}$$

(26)

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}26}(\dot{t)}+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}26}(\dot{t)}={\Phi }_{18}{\mathrm{P}}_{\mathrm{S}5}(\dot{t)}$$

(27 )

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}27}(\dot{t)}+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}27}(\dot{t)}={\Phi }_{19}{\mathrm{P}}_{\mathrm{S}5}(\dot{t)}$$

(28 )

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}28}(\dot{t)}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}28}(\dot{t)}={\Phi }_{20}{\mathrm{P}}_{\mathrm{S}5}(\dot{t)}$$

(29 )

$${\mathrm{P{\mathrm{^{\prime}}}}}_{\mathrm{S}29}(\dot{t)}+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}29}(\dot{t)}={\Phi }_{21}{\mathrm{P}}_{\mathrm{S}5}(\dot{t)}$$

(30)

Equations 31 to 60

$$\left[{\Phi }_{17}+{\Phi }_{18}+{\Phi }_{19}+{\Phi }_{20}+{\Phi }_{21}\right]{\mathrm{P}}_{\mathrm{S}0}\left(\mathrm{t}\right)={\upmu }_{17}{\mathrm{P}}_{\mathrm{S}1}\left(\mathrm{t}\right)+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}2}\left(\mathrm{t}\right)+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}3}\left(\mathrm{t}\right)+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}4}\left(\mathrm{t}\right)+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}5}\left(\mathrm{t}\right)$$

(31)

$$[{\Phi }_{17}+{\Phi }_{18}+{\Phi }_{19}+{\Phi }_{20}+{\Phi }_{21}+{\upmu }_{17}] {\mathrm{P}}_{\mathrm{S}1}\left(\mathrm{t}\right)= {\upmu }_{17}{\mathrm{P}}_{\mathrm{S}6}\left(\mathrm{t}\right)+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}7}\left(\mathrm{t}\right)+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}8}\left(\mathrm{t}\right)+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}9}\left(\mathrm{t}\right) +{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}10}\left(\mathrm{t}\right)+{\Phi }_{17}{\mathrm{P}}_{\mathrm{S}0}\left(\mathrm{t}\right)$$

(32)

$$[{\Phi }_{17}+{\Phi }_{18}+{\Phi }_{19}+{\Phi }_{20}+{\Phi }_{21}+{\upmu }_{18}{]\mathrm{ P}}_{\mathrm{S}2}\left(\mathrm{t}\right)={\upmu }_{17}{\mathrm{P}}_{\mathrm{S}11}\left(\mathrm{t}\right)+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}12}\left(\mathrm{t}\right)+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}13}\left(\mathrm{t}\right)+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}14}\left(\mathrm{t}\right)+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}15}\left(\mathrm{t}\right)+{\Phi }_{18}{\mathrm{P}}_{\mathrm{S}0}\left(\mathrm{t}\right)$$

(33)

$$\left[{\Phi }_{17}+{\Phi }_{18}+{\Phi }_{19}+{\Phi }_{20}+{\Phi }_{21}+{\upmu }_{19}\right]{\mathrm{P}}_{\mathrm{S}3}\left(\mathrm{t}\right)={\upmu }_{17}{\mathrm{P}}_{\mathrm{S}16}\left(\mathrm{t}\right)+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}17}\left(\mathrm{t}\right)+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}18}\left(\mathrm{t}\right)+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}19}\left(\mathrm{t}\right)+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}20}\left(\mathrm{t}\right)+{\Phi }_{19}{\mathrm{P}}_{\mathrm{S}0}\left(\mathrm{t}\right)$$

(34)

$$[{\Phi }_{17}+{\Phi }_{18}+{\Phi }_{19}+{\Phi }_{20}+{\Phi }_{21}+{\upmu }_{20}{]\mathrm{ P}}_{\mathrm{S}4}\left(\mathrm{t}\right)={\upmu }_{17}{\mathrm{P}}_{\mathrm{S}21}\left(\mathrm{t}\right)+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}22}\left(\mathrm{t}\right)+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}23}\left(\mathrm{t}\right)+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}5}\left(\mathrm{t}\right)+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}24}\left(\mathrm{t}\right)+{\Phi }_{20}{\mathrm{P}}_{\mathrm{S}0}\left(\mathrm{t}\right)$$

(35)

$${[\Phi }_{17}+{\Phi }_{18}+{\Phi }_{19}+{\Phi }_{20}+{\Phi }_{21}+{\upmu }_{20}]{\mathrm{P}}_{\mathrm{S}5}={\upmu }_{17}{\mathrm{P}}_{\mathrm{S}25}+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}26}+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}27}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}28}+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}29}+{\Phi }_{20}{\mathrm{P}}_{\mathrm{S}4}$$

(36)

$${\upmu }_{17}{\mathrm{P}}_{\mathrm{S}6}\left(\mathrm{t}\right) ={\Phi }_{17}{\mathrm{P}}_{\mathrm{S}1}\left(\mathrm{t}\right)$$

(37)

$${\upmu }_{18}{\mathrm{P}}_{\mathrm{S}7}\left(\mathrm{t}\right)={\Phi }_{14}{\mathrm{P}}_{\mathrm{S}0}\left(\mathrm{t}\right)$$

(38)

$${\upmu }_{19}{\mathrm{P}}_{\mathrm{S}8}\left(\mathrm{t}\right)={\Phi }_{19}{\mathrm{P}}_{\mathrm{S}1}\left(\mathrm{t}\right)$$

(39)

$${\upmu }_{20}{\mathrm{P}}_{\mathrm{S}9}\left(\mathrm{t}\right)={\Phi }_{20}{\mathrm{P}}_{\mathrm{S}1}\left(\mathrm{t}\right)$$

(40)

$${\upmu }_{21}{\mathrm{P}}_{\mathrm{S}10}\left(\mathrm{t}\right)={\Phi }_{21}{\mathrm{P}}_{\mathrm{S}1}\left(\mathrm{t}\right)$$

(41)

$${\upmu }_{17}{\mathrm{P}}_{\mathrm{S}11}\left(\mathrm{t}\right)={\Phi }_{17}{\mathrm{P}}_{\mathrm{S}2}\left(\mathrm{t}\right)$$

(42)

$${\upmu }_{18}{\mathrm{P}}_{\mathrm{S}12}\left(\mathrm{t}\right)={\Phi }_{18}{\mathrm{P}}_{\mathrm{S}2}\left(\mathrm{t}\right)$$

(43)

$${\upmu }_{19}{\mathrm{P}}_{\mathrm{S}13}\left(\mathrm{t}\right)={\Phi }_{19}{\mathrm{P}}_{\mathrm{S}2}\left(\mathrm{t}\right)$$

(44)

$${\upmu }_{20}{\mathrm{P}}_{\mathrm{S}14}\left(\mathrm{t}\right)={\Phi }_{20}{\mathrm{P}}_{\mathrm{S}2}\left(\mathrm{t}\right)$$

(45)

$${\upmu }_{21}{\mathrm{P}}_{\mathrm{S}15}\left(\mathrm{t}\right)={\Phi }_{21}{\mathrm{P}}_{\mathrm{S}2}\left(\mathrm{t}\right)$$

(46)

$${\upmu }_{17}{\mathrm{P}}_{\mathrm{S}16}\left(\mathrm{t}\right)={\Phi }_{17}{\mathrm{P}}_{\mathrm{S}3}\left(\mathrm{t}\right)$$

(47)

$${\upmu }_{18}{\mathrm{P}}_{\mathrm{S}17}\left(\mathrm{t}\right)={\Phi }_{18}{\mathrm{P}}_{\mathrm{S}3}\left(\mathrm{t}\right)$$

(48)

$${\upmu }_{19}{\mathrm{P}}_{\mathrm{S}18}\left(\mathrm{t}\right)={\Phi }_{19}{\mathrm{P}}_{\mathrm{S}3}\left(\mathrm{t}\right)$$

(49)

$${\upmu }_{20}{\mathrm{P}}_{\mathrm{S}19}\left(\mathrm{t}\right)={\Phi }_{20}{\mathrm{P}}_{\mathrm{S}3}\left(\mathrm{t}\right)$$

(50)

$${\upmu }_{21}{\mathrm{P}}_{\mathrm{S}20}\left(\mathrm{t}\right)={\Phi }_{21}{\mathrm{P}}_{\mathrm{S}3}\left(\mathrm{t}\right)$$

(51)

$${\upmu }_{17}{\mathrm{P}}_{\mathrm{S}21}\left(\mathrm{t}\right)={\Phi }_{17}{\mathrm{P}}_{\mathrm{S}4}\left(\mathrm{t}\right)$$

(52)

$${\upmu }_{18}{\mathrm{P}}_{\mathrm{S}22}\left(\mathrm{t}\right)={\Phi }_{18}{\mathrm{P}}_{\mathrm{S}4}\left(\mathrm{t}\right)$$

(53)

$${\upmu }_{19}{\mathrm{P}}_{\mathrm{S}23}\left(\mathrm{t}\right)={\Phi }_{19}{\mathrm{P}}_{\mathrm{S}4}\left(\mathrm{t}\right)$$

(54)

$${\upmu }_{21}{\mathrm{P}}_{\mathrm{S}24}\left(\mathrm{t}\right)={\Phi }_{21}{\mathrm{P}}_{\mathrm{S}4}\left(\mathrm{t}\right)$$

(55)

$${\upmu }_{17}{\mathrm{P}}_{\mathrm{S}25}\left(\mathrm{t}\right)={\Phi }_{17}{\mathrm{P}}_{\mathrm{S}5}\left(\mathrm{t}\right)$$

(56)

$${\upmu }_{18}{\mathrm{P}}_{\mathrm{S}26}\left(\mathrm{t}\right)={\Phi }_{18}{\mathrm{P}}_{\mathrm{S}5}\left(\mathrm{t}\right)$$

(57)

$${\upmu }_{19}{\mathrm{P}}_{\mathrm{S}27}\left(\mathrm{t}\right)={\Phi }_{19}{\mathrm{P}}_{\mathrm{S}5}\left(\mathrm{t}\right)$$

(58)

$${\upmu }_{20}{\mathrm{P}}_{\mathrm{S}28}\left(\mathrm{t}\right)={\Phi }_{20}{\mathrm{P}}_{5\mathrm{S}}\left(\mathrm{t}\right)$$

(59)

$${\upmu }_{21}{\mathrm{P}}_{\mathrm{S}29}\left(\mathrm{t}\right)={\Phi }_{21}{\mathrm{P}}_{\mathrm{S}5}\left(\mathrm{t}\right)$$

(60)