Stochastic analysis of edible oil refinery industry

Introduction

A number of researchers have analyzed availability parameters of various industrial systems using different methods. In this research paper sensitivity analysis of Edible Oil Refinery Industry has been discussed using regenerative point graphical technique. Edible oil refinery have four units, subsystem 'A' have sub components in parallel, hence when some of its components fail, the unit 'A' can work in reduced capacity, hence the whole system works in reduced state which fails completely on the failure of unit A. Subunits B, D & E have a component in series so if any of the unit fails than the system is down i.e. failed. Fuzzy logic is used to declare the failure of a unit. All units have distinct failure/repair rates in all states. All 6. The system is discussed for steady-state conditions. 7. Assuming that DRR are never failed.

(𝑖 𝑠 𝑟 → 𝑗): r-th directed simple path from i-state to j-state; r takes positive integral values for different paths from i-state to j-state.

(𝜉 𝑠𝑓𝑓 → 𝑖): A directed simple failure free path from -state to i-state.

Vm,m

: Probability factor of the state m reachable from the terminal state m of the M-cycle. 𝑉 𝑚,𝑚 ̅̅̅̅̅̅: Probability factor of the state m reachable from the terminal state m of the 𝑚 -𝑐𝑦𝑐𝑙𝑒 ̅̅̅̅̅̅̅̅ .

Ri (t): Reliability of the system at time t, given that the system entered the un-failed Regenerative state 'i' at t = 0. Ai (t): Probability of the system in up time at Time 't', given that the system entered Regenerative state 'i' at t = 0. Bi (t): Reliability that the server is busy for doing a particulars job at Time 't'; given That the system entered regenerative state 'i' at t = 0. Vi (t): The expected no. of server visits for doing a job in (o,t] given that the system Entered regenerative state 'i' at t = 0. ',': Dash denotes derivative μi: Mean sojourn time spent in state i, before visiting any other states;

: The total un-conditional time spent before transiting to any other regenerative states, given that the system entered regenerative state 'i' at t=0. ni: Expected waiting time spent while doing a given job, given that the system entered regenerative state 'i' at t=0; ξ: Base state of the system. fj: Fuzziness measure of the j-state.

Full Capacity Working State

Failed State A/a: Unit in full capacity working state / failed state, similarly for other units. wi/ λi: Denote repair failure rates of units Taking into consideration the above assumptions and notations the Transition Diagram of the system is given in Figure 1.

The four units are symbolized as Crusher unit = A, Filter and refining unit = B Boiler and kettle unit = D, Neutralizer unit = E Table 1: Primary, Secondary and Tertiary Circuits associated with the system are given in Vertex i CL1 CL2 0 (0,1,0) (0,3,0),(0,4,0),(0,7,0) (0,1,5,0)

(1,8,1),(1,2,1),(1,6,1) -(1,6,1),(1,2,1),(1,8,1) 1 (1,0,1) (1,8,1),(1,2,1), (1,6,1) (1,5,0,1) (0,4,0),(0,3,0),(0,7,0) -(0,3,0),(0,4,0),(0,7,0) 2

(2,1,2) (1,8,1),(1,6,1),(1,0,1),(1,5,0,

(1,8,1),(1,6,1),(1,5,0,1),(1,2,1) (1,0,1) 4 (4,0,4) (0,1,0),(0,3,0),(0,7,0),(0,1,5,0) 5 (5,0,1,5) (0,1,0),(0,3,0),(0,4,0),(0,7,0) (1,0,1), (

→ 1): (0,1) (1,8,1),(1,2,1),(1,6,1)

𝑅 8 (𝑡) = 𝑒 -𝑤 4 µ8 = 1/w4 Mean Sojourn Times Ri(t): Reliability of the system at time t, given that the system in regenerative state i.

: Mean sojourn time spent in state i, before visiting any other states;

Evaluation of Parameters: -The Mean time to system failure and all the key parameters of the system under steady state conditions are evaluated, applying Regenerative Point Graphical Technique (RPGT) and using '0' as the base-state of the system as under:

The transition probability factors of all the reachable states from the base state 'ξ' = '0' are: Probabilities from state '0' to different vertices are given as

Server's Busy Period (B0):

The states where the server is busy are 'j' = 1,2,3,4,5,6,7,8 and regenerative states are 'i' = 0 to 8, taking ξ = '0', using RPGT is given as

B0=1-V0,0μ0/1/(λ1+λ3+λ4+λ5)+{1+(λ5/w5)+(λ3/w3)+(λ4/w4)}+{ λ1(w1+λ2+λ3+λ4+λ5) /(λ1+λ3+λ4+λ5)(w1+λ2+λ3+λ5)(w1+λ2+λ3+λ4)(w1+λ2+λ4+λ5)+(λ4/w4)+(λ5/w5)+(λ2/w2) +(λ3/w3)

Expected Number of Inspections by the repair man:

The regenerative states where the repair man does this job are j = 2,3,4,5,6,7,8, Taking 'ξ' = '0', the number of visit by the repair man is given by

Sensitivity Analysis w.r.t. change in repair rates Fixing λi=0.10 (1≤ 𝑖 ≤ 5) and varying w1, w2, w3, w4, w5 one by one respectively at 0.80, 085, 0.90, 0.95, 1 The Mean time of system failure is fairly very large i.e. 3.2 and is independent of repair rates of units. Practically busy period of the system should decrease with increase in repair rates of units which is shown by the table and graph. The busy period is minimum when repair rate of system 'A' is 1 and minimum value is 0.30597 busy period of the server is maximum when Boiler and kettle unit repair rate in comparison to other units hence repairman should be efficient in repairing the Boiler and kettle unit. If the repair rates are lowest than the server will have to be more busy for fixing the problem, hence server will take more time to repair the units from the above table and graph while observing from top to bottom in columns we see that busy period of server decreases which is the practical trend in almost in all simulation above graph also verify for the similar results. The above table shows that the busy period of the server decreases with the increase in failure rates of units and is minimum hen the failure rate of unit E is minimum in comparison to the failure rates of other units. There is no significant change in the value of the expected fractional number of inspection due to increase in the values of the failure rates of the units but is minimum when the failure rate or unit E is minimum in comparison to the failure rates of other units.

Availability of the Systems (A0): -For various repair rates availability table is

Expected Fractional

Expected Fractional Number of Inspection by the Repairman (V0) -For different values of failure rate, the corresponding table is given below

Conclusion

To have optimum vale of system parameters management may control the failure and repair rates of units depending upon the availability of finances and market circumstances. Any state can be taken as the Base-state to evaluate the various parameters. Study can also be extended for time dependent cases also.

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