To Calculate the Alternate Formula for Production Rate When the When the Degree of Imbalance (D) in A Un-Paced Production Line

INTRODUCTION

In revaluation of Industrial development, All the Industries are Automatic System Plants. They utilize the CIM (Computer Integrated Manufacturing) or Flexible Manufacturing system for assembling the machine parts to make useful goods (Finished job). When we do the assembled the parts the time are not same for all work station because some component take more time some components take less time to Assemble it (e.g. when assemble a car steering system take more time rather than assembly of chassis or wheels).Due to variation in work load distribution (time) in each station are different the pacing agencies in production line are called un-paced production line. Calculating the production rate Up to 5-workstation is very difficult task even in computer programming (Jamali, et. al., 2015) therefore we develop an easy formula (Jamali) for production rate and say it alternate formula and also validate it from -3 σ to + 3 σ. Now we use all 60 outcomes (work load distribution‖ time‖ in work station) found in our study (Jamali, 2002) in production rate 5-workstation for un-paced production line and generate the near about exact Alternate formula (AFR) for production rate. And these formulas are easy to calculate higher (more than 5- work station) work station production rate.

Here we Know the interval of time from previously we find out the actual production rate Stretegies (Jamali, 2015. Hiller and So, 1993. Jamali & Prof. Suhail, 2016. Jamali, Prof. Suhail, 2016. Jamali, 2002. Jamali, 2002) When we know we the Correction Factor, variance with mean value of Correction Factor (Σ CFi/n-CFi), standard deviation σ then we find the actual formula for production rate whose utilize in to calculating higher order workstation in production line. We define here each function. Here D(degree of imbalance)= Highest work load (Time) distribution in work station – Lowest work load (time) distribution in work station Correction Factor = Production Rate (R) - Alternate formula for Production Rate (AFR) Alternate formula Rate = (μ1 + μ 5)/ total time in production line Here μ1, μ2, μ3, μ4, μ5 is work load (time) distribution on work station 1,2,3,4,5 Mean value of Correction Factor = Σ CF/n

rate) (Jamali, 2015, 2002). For example, for S. No.1, 2, 3, ………..30,31……… 59, 60 means Σ CF/n – CF1, Σ CF/n – CF2, Σ CF/n – CF5……., Σ CF/n – CF30, Σ CF/n – CF31………….. Σ CF/n – CF55, Σ CF/n – CF60 Where standard deviation σ = Sqrt (Σ (Σ CF/n- CFi)2)/n Here i indicate that S. No. and n =total S.No.= 60 For example, for S. No.1, 2,……….30 ,31……………… 59, 60 means Sum (S) = (Σ CF/n- CF1)2 + (Σ CF/n- CF3)2 +………………………+(Σ CF/n-CF30)2+(Σ CF/n-CF31)2+………………….(Σ CF/n- CF59)2+(Σ CF/n- CF60)2 In the below tables we found the Production Rate (Jamali, 2015, 2002). AFR (Alternate Formula Rate), Correction Factor, Variance with Mean vale of Correction Factor (Σ CF/n-CFi) and standard deviation σ = Sqrt (Σ (Σ CF/n- CFi)2)/n in D= 1.2 and D=0.4 linear work load distribution. And then we found the corrected Alternate Formula for production Rate (AFPR) Therefore we say( found) that AFPR = AFR ± (a +bi X i-1) σ Here we say that for work load distribution Means>1 we take -Ve sign upto Sr. No.(n/4) and some values up to mid Sr. no Because the we take only work load distribution (First and Last work station and ignore the other work load distribution in work station )and after words mid Sr. no we take +Ve Sign of σ the If Workloads Distribution < 1 we take + Ve sign for all values of σ because the work load distribution on work station difference are not more.

D=1.2 linear, the values of CFi(-0.1111, -0.11232 … -0.0056 10… -0.009215 …… 0.045120, ….. 0.042125 …….. -0.023330 …………… 0.091435, ……. 0.088740 …… 0.084345 …. 0.080850, ……… 0.132555 ……… 0.193160) σ = 0.14234 values of D=0.4 linear, the values of CFi (0.03071, 0.03022 ……… 0.02925 ……. 0.045410 …….. 0.062720 ……. 0.059430 …… 0.07640 ………… 0.113650 ….. 0.128760) σ = 0.00425 Here we know that the formula of production rate are AFPR = AFR ± (a +b) σ Where for D= 1.2 linear a = CF/ σ = -0.111/ 0.14234= -0.7798 for S.No.1 that‘s be nearly equal to a= D/2 + (D-1) = (1.2 /2) + (1.2-1) = 0.6+0.2= 0.8 for b = (CF1 - CF2)/ σ = (- 0.111- (-0.1123))/0.14234 = -0.009133 ≈ 0.01 for S. NO. 2 and onwards upto 30 because Upto 30 values are –ve Sign, for S. No. 1 the value of b= 0 then we calculate the + Ve values are b= (CF20- CF21)/ σ = (0.0451-0.0437)/0.14234 =0.00983 ≈ 0.1 therefore we say that the value of b = 0.1 X S. No. (i-1) where i= 1 to 60 we calculate the last value (Worst production Rate) the formula is AFPR = AFR ± (a + b) σ For Example S. No.60 the production Rate is(here S. No. is higher than 30 then we take +Ve) hence AFPR= (AFR 60 + (a + bi-1) σ) = 0.22 +(0.8+59 X0.01) X0.14234 = 0.4178 ≈ 0.4131 (R60) For Example S. No 2 AFPR = AFR 60 + (a + bi-1) σ = 0.58 – (0.8 - 0.01 x1)x 0.14234=0.4675 ≈ 0.4677 (R2) Therefore our formula is correct when we know the D (degree of Imbalance) for higher values of work load distribution in production rate in production line Similarly for Where for D=0.4 linear. All the values are + Ve here in formula therefore we calculate a and b a= CF1/ σ = 0.0307/ 0.033 = 0.93 = (3D/2+3D/4) = ((3x0.4/2) + (3x0.4/4)) = 0.9 here b1 = 0 and b2= (CF1 – CF2)/ σ =( 0.0307 -0.0302)/0.033 ≈ 0.015 b20 = (CF20 – CF21)/ σ = (0.0627-0.0624)/ 0.033 ≈ 0.0091

therefore we take b i-1 = 0.01 for each S.NO. ( i-1) where σ = D 0.4/10 = 0.04

Therefore we conclude that our formula 1. AFPR = AFR 60 + (a + bi-1) σ is applicable when we Know that when (a= 3D/2+3D/4)) where D0.4 =0.4, b i-1= 0.1X S.NO. (i-1) and σ = D 0.4/10 for each lower values of degree of imbalance (D<1) in a production line 2. AFPR = AFR ± (a +b) σ is applicable when we Know that When (a= D/10+( D-1)/10) where D=D , b i-1= 0.1X S.NO. (i-1) and σ = D /10 X 0.15 for each higher values of degree of imbalance (D>1) in a production line 3. Formula is valid because in both graph we say that the Variance with Mean vale Of Correction Factor (-0.15 to +0.15 for D=1.2 and -0.05 to -0.05for D= 0.4) is near about -3 σ to +3 σ when the Correction factor is not included. 4. These strategies (formula) are very useful when we calculate production rate the workload distribution higher than 5-work station (like 7- 9-11 –or more work station) in production line

We make a Simple program in C to calculate the production rate of 7-and analyze the results and also calculate the production rate when the Setup time on work station and work load distribution known in batch type production rate in Flexible manufacturing system or Computer Integrated Manufacturing.

1. M. T. Jamali, Mohd Ziaulhaq, Dr. A. Suhail, Mohammad Khalid (Jan- Feb. 2015). ―The Effect of Imbalance on the Performance of Un-paced Production Line – A Mathematical Modeling Approach‖, IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE), PP 14-19, Volume 12, Issue 1 Ver. I, www.iosrjournals.org 2. Hiller, F.S. and K.C. So (1993). some data for applying the bowel phenomenon to large production line system ―int. J.prod. res., 31,811 known‖ www. academia.com‖ 4. M. T. Jamali, Prof. Arif Suhail (2016). ―THE EFFECT OF IMBALANCE ON THE PERFORMANCE OF UN-PACED PRODUCTION LINE WHEN THE OPTIMAL BOWL UNKNOWN‖ International journal of core Engineering and management (ISSN 2348-9510, vol. 3, issue 8, Nov 2016), www.ijcem.in 5. Mohammad Tarique Jamali ―To find the alternate formula for production rate trends to design for un-paced production line when unbalancing the workloads.‖ Send to IMA chief academic officer 6. Mohammad Tarique Jamali ―To calculate the validity of alternate formula for production rate to design for un-paced production line when unbalancing the workloads.‖ Send to IMA chief academic officer 7. Kunter S. Akbay (1996). ―Using simulation optimization to find the best solution‖, Ind. Engg. Solution 28 No. 5 8. H. A. Beg (1996). ―A study of performance of Production line ―. M.Tech. Dissertation, AMU Aligarh 9. Mohammad Tarique Jamali (2002). ―The effect of performance of imbalance on the performance of Unpaced production line‖ M.Tech dissertation, AMU Aligarh, India

Research Scholar, Young Scientist University, USA (IMA-INT-17-19 ADC-271051) and Faculty, Department of Industrial Engineering, King Khalid University, Abha, Kingdom of Saudi Arabia