Data model development for addressing problem of Linxens - removing reels with Punches

1. Input from Customer (Essentially columns A-M in the step sheet, to see from Row 16 Onwards no. of rows could be varying for different items)

CSV file for tracks in our example

2. Variables and Constant definitions for the optimization problem(i) Number of Punches in a meter These will be constants for our problem.Track A a1, a2, a3, a4., an where n is the number of meters (in this case 275) [In our attached excel file these are values in the cell B17:B291 ; ie a1 = 32, a2 = 8, a3 = 6 etc]Track B - b1, b2, b3, b4., bn [In our attached excel file these are values in the cell C17:C291 ; ie b1 = 7, b2 = 11, b3 = 6 etc]

Track C - c1, c2, c3, c4., cn [In our attached excel file these are values in the cell D17:D291 ; ie c1 = 22,c2 = 18, c3 = 14 etc]

Track D - d1, d2, d3, d4., dn [In our attached excel file these are values in the cell E17:E291 ; ie d1 = 25,d2 = 11, d3 = 6 etc]

(ii) Flag to identify if a particular meter must be removed for a track

This is the set of variables to be observed by usTrack A w1, w2, w3, w4., wn where n is the number of meters (in this case 275) Track B - x1, x2, x3, x4., xn Track C - y1, y2, y3, y4., yn Track D - z1, z2, z3, z4., zn

[New variables / Constants added on 19-Dec-2014]

(iii) Flag to identify if there is an existing Splice in a particular meter (sort of joint/separator because of a prior cut) in a trackTrack A sw1, sw2, sw3, sw4,, swn where n is the number of meters (in this case 275)Track B sx1, sx2, sx3, sx4,.,sxn Track C sy1, sy2, sy3, sy4,synTrack D sz1, sz2, sz3, sz4,szn

If sw1 = 1 it means there is a splice in the first meter of Track A.

(iv) e Error % that is threshold for each track (we assume the threshold is common for all tracks) Depends on the item being produced (could be 2% or 5%)Comment by Sundaresan, Narayanan: Changed from 2% to item based after discussion with client on 19-Dec-2014(v) c Factor input for no. of connectors per 1 meter length (140.35 in our example)(vi) Ms Maximum number of splices allowed in a track (For example if 10 is the maximum number of splices allowed and if 3 is the existing splices in those meters that will not be removed, we should ensure that the track should have no more than 7 new cuts (because every cut will result in a splice))Comment by Sundaresan, Narayanan: Added after discussion with client on 19-Dec-2014(vii) Ming Minimum number of good quantities required in a track after removal of bad meters.(viii) Maxg Maximum number of good quantities required in a track after removal of bad meters.

3. Arunan will work out the first level of removal using specific logic based on columns F to M . He will help us have some values of w, x, y and z (This is columns NQ in the attached excel sheet)In our case, for example , w152=1=w265=w266=w268=w270=w271=w272=w273=w274x32=1=x120=x268=x270=x271=x272=x273=x274y269=1=y270=y272=y273=y274=y275z264=1=z266=z268=z270=z271=z273=z275) 4. Arunan will also give us the first level splice inputs for Track A, B, C and D respectively.Comment by Sundaresan, Narayanan: Added after discussion with client on 19-Dec-2014

5. Below is the first modelling of our function [I reviewed this and in general this should help us solve the problem for the first few iterations ie without moving pieces from Track A to Track B and so on]

We will create problems by each Track and solve them separately I have given below for Track A.

Objective Function(s) [Maximize the number of defective punches to be removed] nMax. PRa = aiwi = a1w1 + a2w2 + ... + anwn i = 1PRa = Punch Removed in A nComment by Sundaresan, Narayanan: To check if this has to be changed to a Maximize objective with a different variable instead of a minimize objective. We can use Good quantity as the variable and use objective function as maximize good quantityMin. LRa = wi = w1 + w2 + ... + wn i = 1[Minimize the length being removed as length has also got good quantity besides punches]

LRa = Length Removed in A

Subject to Constraints(1) 2 wi - wi-1 - wi+1 > -2 for all i = 2n-1 [This will remove the condition where wi is in between 2 punches to be removed ie wi-1 = wi+1 ie; ie resulting in a sequence like 1-0-1] [The requirement is that there should be a min. of 2 meters continuous retained meter not removed, we can have 1-0-0-1 but not 1-1-0-1 for example](2) (a1(1-w1)+ a2(1-w2)+ ... + an(1-wn))/((n-( w1 + w2 + ... + wn)*c) OKQuantity of splices => OKMeters mandatory to be removed => OKLength with minimal splice distance removed => OKDefect rate => NOK for A, B and CQuantity of good parts => OKQuantity of splices => OKBelow, the count of punch it should be necessary to remove in order to reach the "max 2% defect rate"TrackABCDABCDABCDinitial defect rate4.27%4.85%8.30%4.07%new defect rate2.79%3.55%6.97%2.70%2.47%3.28%6.68%1.78%new punch count1042132926301015912122025126581744761760-80new total length266267269268263265268263new good quantity3629136144351243659936176365423692436000new splice count54262211

Length removed98673215initial sum of punch1648187132021570punch removed606542572555130109118357

1 - Lenght must be removed (Spunch)2 - Remove length with splice distance =< 3mmeter removed at this 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