Performance evaluation and optimization for the stock preparation system of a paper plant using genetic algorithm.

Khanduja, Rajiv ; Tewari, P.C. ; Chauhan, R.S. 等

Introduction

The paper industry comprises of large complex engineering systems arranged in series, parallel or a combination of both the configurations. Some of these systems are chipping, cooking, washing, bleaching, screening, stock preparation and paper production etc. These systems are normally arranged in hybrid configuration. The important process of a paper industry, upon which the quality of paper depends, is the stock preparation process. In the process of paper formation, the chips from storage are fed in to a digester to form the pulp, which is processed through various systems called knotter, decker, opener and washing. Then the washed pulp is bleached to get chlorine free white pulp, which is further passed through screen and cleaner to separate out oversize and odd shape particles. After that in the stock preparation refining of pulp, addition of chemicals and filler (to change the paper properties), centri-cleaning and screening of pulp is done before finally sending it to paper making machine.

Literature Review

The available literature reflects that several approaches have been used to analyze the steady state behaviour of various systems. Dhillon et al. (1981) have frequently used the Markovian approach for the availability analysis, using exponential distribution for failure and repair times. Kumar, D. et al. (1988, 1989 and 1993) dealt with reliability, availability and operational behavior analysis for different systems in the paper plant. Kumar et al (1988, 1993) dealt with maintenance planning for the systems in fertilizer and thermal plants. Kalyanmoy Deb (1995) has explained the optimization techniques and how they can be used in the engineering problems. Shooman (1996) suggested different methods for the reliability computations of systems with dependent failures. Sunand et al. (1999) dealt with maintenance management for Ammonia Synthesis System in fertilizer plant. Tewari et al. (2003) dealt with the determination of availability for the systems with elements exhibiting independent failures and repairs or the operation with standby elements for sugar industry. He also dealt with mathematical modeling and behavioral analysis for a refining system of a sugar industry using Genetic Algorithm. Sunand et al. (2007) discussed simulated availability of C[O.sub.2] cooling system in a fertilizer plant. Rajiv et al. (2007) have developed Decision Support System for Stock Preparation System of Paper Plant. He also dealt with availability of bleaching system of paper plant.

System Description

The Stock Preparation system comprises of five main subsystems, which are as follows:

* Chest (A): This subsystem consists of three units in series used to add the sizing chemicals and mixing is done with the agitator. The failure of anyone

* causes the complete failure of the system.

* Refiner (B): This subsystem consists of three units in parallel to crush and brush the fibre. Failure of anyone reduces the capacity of the system and complete failure of the system occurs when all three refiners fail.

* Fan pump (C): This subsystem consists of one unit to mix the stock from refiners with white liquid and china clay. Its failure causes complete failure of the system.

* Centri-cleaner (D): This subsystem consists of one unit to remove the unwanted and dirt particles from the stock by centrifugal force action. Its failure causes complete failure of the system.

* Screen (E): This subsystem consists of one unit to remove foreign particle. Its failure causes complete failure of the system.

Assumptions and Notations

The transition diagram (figure1) of Stock Preparation system shows the two states, the system can acquire i.e. full working and failed state. Based on the transition diagram, a performance-evaluating model has been developed. The assumptions and notations associated with the transition diagram of Stock Preparation system are as follows:

Assumptions

(1) Failure/repair rates are constant over time and statistically independent.

(2) A repaired unit is good as new, performance wise for a specified duration.

(3) Sufficient repair facilities are provided.

(4) Standby units are of the same nature as that of active units.

(5) System may work at reduced capacity.

(6) Service includes repair and/or replacement.

(7) There are no simultaneous failures.

Notations

A,B,C, D,E : represent good working states of respective chest, refiner, fan pump, centri-cleaner, screen.

a,b,c,d,e : represent failed states of respective chest, refiner, fan pump, centri-cleaner, screen.

[[lambda].sub.1],[[lambda].sub.2],[[lambda].sub.3],[[lambda].sub.4], [[lambda].sub.5] : respective mean constant failure rates of A,B C, D,E.

[[micro].sub.1],[[micro].sub.2],[[micro].sub.3],[[micro].sub.4], [[micro].sub.5] : respective mean constant repair rates of a,b,c,d,e.

[P.sub.0](t) : Probability that the system is working at full capacity at time t.

[P.sub.i](t) : Probability that the system is in the ith state at time t.

P': First-order derivative of the probabilities.

[FIGURE 1 OMITTED]

Performance Evaluation

The performance modeling is carried out using simple probabilistic considerations and differential equations are developed on the basis of Markov birth-death process. These equations are further solved for determining the steady state availability of the Stock Preparation system. Various probability considerations give the following differential equations associated with the Stock Preparation system:

State 0, 1, 2--Represents full capacity working with no standby.

State 3 to 15--Represents the system in the failed state.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

With initial conditions at time t = 0

[P.sub.i] (t) = 1 for i = 0

= 0 for [not equal to] 0

Steady state Analysis

Since the paper plant is a process industry, its every unit should be available for long period. Therefore, long run availability of the system is computed by substituting P'[right arrow] 0 as t [right arrow] [infinity] for equations (1)-(8) and solving them recursively;

[P.sub.1] = [B.sub.2] [P.sub.0]

[P.sub.2] = [B.sub.2.sup.2] [P.sub.0]

[P.sub.3] = [B.sub.1] [P.sub.0]

[P.sub.4] = [B.sub.3] [P.sub.0]

[P.sub.5] = [B.sub.4] [P.sub.0]

[P.sub.6] = [B.sub.5] [P.sub.0]

[P.sub.7] = [B.sub.1] [B.sub.2] [P.sub.0]

[P.sub.8] = [B.sub.3] [B.sub.2] [P.sub.0]

[P.sub.9] = [B.sub.4] [B.sub.2] [P.sub.0]

[P.sub.10] = [B.sub.5] [B.sub.2] [P.sub.0]

[P.sub.11]= [B.sub.1] [B.sub.2.sup.2] [P.sub.0]

[P.sub.12] = [B.sub.2.sup.3] [P.sub.0]

[P.sub.13] = [B.sub.3] [B.sub.2.sup.2] [P.sub.0]

[P.sub.14] = [B.sub.4] [B.sub.2.sup.2] [P.sub.0]

[P.sub.15] = [B.sub.5][B.sub.2.sup.2] [P.sub.0]

Where

[B.sub.i] = [lambda]i/[mu]I i= 1,2,3,4,5

Using normalizing condition i.e. sum of all the state probabilities is equal to one i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

The steady state availability (Av.) of this Stock Preparation system is given by summation of all the full working and reduced capacity states.

Av. = [P.sub.0] + [P.sub.1] + [P.sub.2]

Av. = [P.sub.0] + [B.sub.2] [P.sub.0] + [B.sub.2.sup.2] [P.sub.0]

Av. = [P.sub.0][1+[B.sub.2] +[B.sub.2.sup.2]

Availability (Av.) = [1+[B.sub.2]+[B.sub.2.sup.2]/[1+(1+[B.sub.2]+ [B.sub.2.sup.2])([B.sub.1] [B.sub.2] + [B.sub.3] + [B.sub.4] + [B.sub.5])]

Here, system performance has been evaluated in terms of availability.

Genetic Algorithm

Genetic Algorithms are computerized search and optimization algorithms based on the mechanics of natural genetics and natural selection. Genetic Algorithms have become important because they are found to be potential search and optimization techniques for complex engineering optimization problems. The action of Genetic Algorithm is shown in figure-2 for parameter optimization in the present problem can be stated as follows:

(1) Initialize the parameters of the Genetic Algorithm.

(2) Randomly generate the initial population and prepare the coded strings.

(3) Compute the fitness of each individual in the old population.

(4) Form the mating pool from the old population.

(5) Select two parents from the mating pool randomly.

(6) Perform the crossover of the parents to produce two off springs.

(7) Mutate if required.

(8) Place the child strings to new population.

(9) Compute the fitness of each individual in new population.

(10) Create best-fit population from the previous and new population.

(11) Repeat the steps 4 to 10 until the best individuals in new population represent the optimum value of the performance function (Unit Availability).

[FIGURE 2 OMITTED]

Performance Optimization Using Genetic Algorithm

The performance optimization of the Stock Preparation system is highly influenced by the failure and repair parameters of each subsystem. These parameters ensure high performance of the Stock Preparation system. Genetic Algorithm is hereby proposed to coordinate the failure and repair parameters of each subsystem for stable system performance i.e. high availability. Here, number of parameters is ten (five failure parameters and five repair parameters). The design procedure is described as follows:

To use Genetic Algorithm for solving the given problem, the chromosomes are to be coded in real structures. Unlike, unsigned fixed point integer coding parameters are mapped to a specified interval [[X.sub.min], [X.sub.max]], where [X.sub.min] and [X.sub.max] are the minimum and maximum values of system parameters. The maximum value of the availability function corresponds to optimum values of system parameters. These parameters are optimized according to the performance index i.e. desired availability level. To test the proposed method, failure and repair rates are determined simultaneously for optimal value of unit availability. Effect of number of generations and population size on the availability of the Stock Preparation system is shown in Table 1 and 2. To specify the computed simulation more precisely, trial sets are also chosen for Genetic Algorithm and system parameters. The performance [availability] of the Stock

Preparation system is evaluated by using the designed values of the unit parameters. Failure and Repair Rate Parameter Constraints are [[lambda].sub.1], [[micro].sub.1], [[lambda].sub.2], [[micro].sub.2], [[lambda].sub.3], [[micro].sub.3], [[lambda].sub.4], [[micro].sub.4], [[lambda].sub.5], [[micro].sub.5]

[[lambda].sub.1] [epsilon] [0.005, 0.0250] [[lambda].sub.2] [epsilon] [0.01, 0.09] [[lambda].sub.3] [epsilon] [0.01, 0.05] [[lambda].sub.4] [epsilon] [0.008, 0.07] [[lambda].sub.5] [epsilon] [0.01, 0.09]

[[micro].sub.1] [epsilon] [0.05, 0.25] [[micro].sub.2] [epsilon] [0.05, 0.45] [[micro].sub.3] [epsilon] [0.05, 0.25] [[micro].sub.4] [epsilon] [0.10, 0.90] [[micro].sub.5] [epsilon] [0.10, 0.50] Here, real-coded structures are used. The simulation is done to maximum number of generations, which is varying from 20 to 100. The effect of number of generations on availability of the Stock Preparation system is shown in figure3.The optimum value of system's performance is 72.79%, for which the best possible combination of failure and repair rates is [[lambda].sub.1] =0.0083, [[micro].sub.1] =0.2253, [[lambda].sub.2] =0.0706, [[micro].sub.2] =0.3154, [[lambda].sub.3] =0.0104, [[micro].sub.3] =0.1668, [[lambda].sub.4] =0.0095, [[micro].sub.4] =0.4911, [[lambda].sub.5] =0.0374, [[micro].sub.5] =0.2342 at generation size 60 as given in table 1.

Now the simulation is done to maximum number of population size, which is varying from 20 to 100. The effect of population size on availability of the Stock Preparation system is shown in figure4.The optimum value of system's performance is 72.56%, for which the best possible combination of failure and repair rates is [[lambda].sub.1] =0.0051, [[micro].sub.1] =0.2342, [[lambda].sub.2] =0.0254, [[micro].sub.2] =0.2961, [[lambda].sub.3] =0.0103, [[micro].sub.3] =0.2449, [[lambda].sub.4] =0.0098, [[micro].sub.4] =0.8029, [[lambda].sub.5] =0.0541, [[micro].sub.5] =0.2994 at population size 90 as given in table 2.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Conclusions

The performance evaluation and optimization of the Stock Preparation system of a paper plant has been carried out in this paper. Genetic Algorithm is hereby proposed to select the various feasible values of the system failure and repair parameters. Then, Genetic Algorithm is successfully applied to coordinate simultaneously these parameters for an optimum level of system performance. Besides, the effect of Genetic Algorithm parameters such as number of generations and population size on the system performance i.e. availability has also been analysed. Then, the findings of this paper are discussed with the concerned paper plant management. Such results are found highly beneficial for the purpose of performance optimization of a stock preparation system in the paper plant concerned.

References

[1] Dhillon, B.S., Singh, C., 1981, Engineering Reliability- New techniques and applications. John willey and sons, New York.

[2] D.E. Goldberg., 2001 Genetic Algorithm in Search, Optimization and Machine Learning, Pearson Edition. Asia.

[3] Kumar, D., Singh, I.P., and Singh, J., 1988, "Reliability analysis of the Feeding System in the Paper Industry". Microelectron Reliability, 28(2), pp.213-215.

[4] Kumar, Dinesh, Singh, Jai, and Pandey, P.C., 1989, "Availability analysis of the washing system in the paper industry". Microelectron Relaibility, vol.29, pp.775-778.

[5] Kumar, Dinesh, Singh, Jai, and Pandey, P.C., 1993, "Operational behavior and profit function for a bleaching and screening system in the paper industry". Microelectron Relaibility, vol. 33, pp.1101-1105.

[6] Kalyanmoy Deb, 1995, Optimization for Engineering Design: Algorithms and examples, Prentice Hall of India, New Delhi, India.

[7] P.C. Tewari, D. Joshi, M. Sreenivasa Rao., 2005, "Mathematical Modeling and Behavioural Analysis of a Refining System using Genetic Algorithm", Proceedings of National Conference on Competitive Manufacturing Technology & management for Global Marketing, Chennai, pp. 131-134.

[8] Rajiv Khanduja, Tewari, P.C., Dinesh Kumar., 2008, "Development of Performance Evaluation System for Screening Unit of Paper Plant". International Journal of Applied Engineering Research, Vol.3, Number 3, pp. 451-460.

[9] Rajiv Khanduja, Tewari, P.C., Dinesh Kumar., 2008, "Availability Analysis of Bleaching System of Paper Plant." Journal of Industrial Engineering, Udyog Pragati, N.I.T.I.E. Mumbai (India), 32(1), pp.24-29.

[10] Srinath, L.S., (1994), Reliability Engineering. 3rd edition, East-West Press Pvt. Ltd., New Delhi, India.

[11] Shooman, M.L., 1996, "Reliability Computation for Systems with Dependents Failures". Proceedings of IEEE Annual Symposium on Reliability, pp.44-56.

[12] Sunand Kumar, Dinesh Kumar, Mehta, N.P., 1999, "Maintenance Management for Ammonia Synthesis System in a Urea Fertilizer Plant". International Journal of Management and System (IJOMAS), 15(3), pp.211-214.

[13] Sunand Kumar, Tewari, P.C., Sharma Rajiv., 2007, Simulated Availability of C[O.sub.2] Cooling System in a Fertilizer Plant. Industrial Engineering Journal (Indian Institution of Industrial Engineering, Mumbai), 36(10), pp.19-23.

[14] Tewari, P.C., Kumar, D., Mehta, N.P., 2000, Decision Support System of Refining System of Sugar Plant. Journal of Institution of Engineers (India), 84, pp. 41-44.

Er. Rajiv Khanduja (1) P.C. Tewari (2) and Er. R.S. Chauhan (3)

(1) Asstt. Professor, Department of Mechanical Engineering, SKIET, Kurukshetra136118, Haryana,India. E-mail : [email protected]

(2) Asstt. Professor, Department of Mechanical Engineering, NIT, Kurukshetra-136119, Haryana,India. E-mail : [email protected]

(3) Asstt. Professor, Department of Electronic and Communication Engineering, JMIT, Radaur, Yamuna Nagar-135133, Haryana, India. E-mail: [email protected]

Table 1: Effect of Number of Generations on Availability of the Stock
Preparation System Using Genetic Algorithm.
(Mutation Probability = 0.015, Population Size = 80, Crossover
Probability = 0.85).

Number of
Generations   Availability   [lambda.sub.1]   [mu.sub.1]

20                  0.6939           0.0052       0.2019

30                  0.7138           0.0053       0.2401

40                  0.7243           0.0051       0.2314

50                  0.7246           0.0059       0.2414

60                  0.7279           0.0083       0.2253

70                  0.7194            0.005       0.2340

80                  0.7178           0.0051       0.1855

90                  0.7165           0.0054       0.2366

100                 0.7164           0.0054       0.2500

Number of
Generations   [lambda.sub.2]   [mu.sub.2]   [lambda.sub.3]   [mu.sub.3]

20                    0.0402       0.4153           0.0109       0.2142

30                    0.0111       0.2730           0.0104       0.2450

40                    0.0426       0.3923           0.0105       0.2413

50                    0.0232         0.45             0.01         0.25

60                    0.0706       0.3154           0.0104       0.1668

70                    0.0103         0.45           0.0107       0.2471

80                    0.0444       0.3910           0.0117       0.2350

90                    0.0379       0.1774             0.01       0.2437

100                   0.0118         0.45           0.0107       0.2364

Number of
Generations   [lambda.sub.4]   [mu.sub.4]   [lambda.sub.5]   [mu.sub.5]

20                    0.0143       0.6482           0.0759       0.2763

30                     0.008       0.7665             0.01       0.2966

40                    0.0132       0.7442           0.0717       0.3131

50                     0.008         0.90           0.0524       0.4842

60                    0.0095       0.4911           0.0374       0.2342

70                     0.008       0.8814           0.0136       0.2265

80                    0.0166       0.6957           0.0591       0.1366

90                    0.0093       0.8875           0.0674       0.4299

100                   0.0101       0.8307           0.0509       0.3505

Table 2: Effect of Population Size on Availability of the Stock
Preparation System Using Genetic Algorithm. (Mutation
Probability=0.015, Number of Generations=80, Crossover
Probability=0.85).

Population
Size         Availability   [lambda.sub.1]   [mu.sub.1]

20                 0.7028           0.0111       0.1950

30                 0.7156            0.005       0.2018

40                 0.7236           0.0058       0.1873

50                 0.7145            0.005       0.2500

60                 0.7169           0.0052       0.2304

70                 0.7210           0.0099       0.2457

80                 0.7252           0.0052       0.2270

90                 0.7256           0.0051       0.2342

100                0.7252           0.0054       0.2448

Population
Size         [lambda.sub.2]   [mu.sub.2]   [lambda.sub.3]   [mu.sub.3]

20                   0.0635       0.2322           0.0108       0.1415

30                   0.0124       0.2281             0.01       0.2500

40                   0.0278       0.4238           0.0102       0.2500

50                   0.0196       0.3779             0.01       0.2254

60                   0.0118       0.3914             0.01       0.2422

70                   0.0205       0.1093           0.0101       0.2453

80                   0.0352       0.3842             0.01       0.2500

90                   0.0254       0.2961           0.0103       0.2449

100                  0.0354       0.3845             0.01       0.2460

Population
Size         [lambda.sub.4]   [mu.sub.4]   [lambda.sub.5]   [mu.sub.5]

20                   0.0161       0.5048           0.0447       0.1826

30                    0.008          0.9           0.0758       0.1829

40                   0.0195       0.7411           0.0742       0.1750

50                   0.0096       0.6805           0.0873       0.3685

60                   0.0088       0.8810           0.0515       0.4543

70                   0.0132       0.6538            0.063       0.2672

80                    0.008       0.7899           0.0195       0.3167

90                   0.0098       0.8029           0.0541       0.2994

100                  0.0126       0.7674           0.0627       0.2483