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by: Tod Murray


Tod Murray
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Class Notes
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This 19 page Class Notes was uploaded by Tod Murray on Friday September 18, 2015. The Class Notes belongs to EIN 6392 at University of Florida taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/206891/ein-6392-university-of-florida in Industrial Engineering at University of Florida.

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Date Created: 09/18/15
FACTORY PHYSICS The term factory physics denotes those manufacturing relationships that can be described mathematically much as the term physics refer to physical relationships that can be framed in mathematical terms In order to understand the factory physics and establish the mathematics needed for tackling production related problems we must first understand the factory dynamics First we provide some definitions Workstation Collection of one or more machines or workers performing identical functions Raw Material parts purchased from outside Components individual pieces that are assembled into more compleX products Subassemblies assembly of several components End items a part that is directly sold to the customer Bill of Materials list of parts that go into a product Consumables items used in production but are not part of a product Routing description of the sequence of workstations passed through by a part Orders request from a customer for a particular part number Purchase order a collection of one or more orders Jobs a set of physical materials that traverse routings together with the logical information Throughput TH the average output of a production process Capacity of a station upper limit in its throughput Crib inventory intermediate inventory location usually at the end of a routing Finished goods inventory place where the end items are held prior to shipping Work in process WIP the inventory between the start and the end points of a production routing All the products between but not including the stock points Cycle time CT average time from release of a job at the beginning of a routing to until it reaches an inventory point at the end of the routing Ie the time the part spends as WIP It is more difficult to define this for the entire product Lead time time allotted for production of a part on the routing Service level Pcycle time lt lead time Fill rate fraction of orders that are filled from stock Bottleneck rate rb rate of the process center with the least long term capacity Raw process time T0 sum of the longterm average process times of each workstation in the line Congestion coefficient at a dimensionless coefficient that measures congestion in the line A SAMPLE FOURSTATION PRODUCTION LINE Factory 1 Consider a production line composed of four machines Identical jobs go through machines 14 repetitively After each machine processes a job it becomes immediately available at the next station for processing The line runs 24 hours a day with breaks and lunches covered by spare operators For the sake of completeness we assume that there is unlimited demand for the product Also assume that each machine takes 2 hours to process a single job From the information given above the capacity or rate of each machine is the same and equals one unit every two hours or 12 parts per hour Therefore from the definition of bottleneck any of the four machines can be regarded as a bottleneck and rb 05 unitshour or 12 units per day In assembly line terminology this is a balanced line since all station have the same capacity The total amount of processing time for one unit through this line is To 24 8 hours Therefore the critical WIP level is given by W0 rbT0 058 4 units Note that W0 is equal to the number of machines in the line Also note that since there is no variability in the system the Congestion Coefficient for the line is OL 0 Factory 2 This is the same four step operations as in Factory lbut now each operation has differing number of machines available and their processing times are different as shown in table below Station of machines Process time Station rate 1 l 2 hr 05 jobshr 2 2 5 hr 04 jobshr 3 6 10 hr 06jobshr 4 2 3 hr 067jobshr The rate of a station consisting of several identical machines in parallel is calculated as the individual machine rate times the number of machines For example the rate per machine at station 3 is 110 jobs per hour so the rate ofthe station is 6110 06 jobs per hour The capacity of the line with multimachine stations is still defined by the rate of the bottleneck or slowest station in the line i Factory2 the bottleneck is station 2 therefore rb 04 jobs per hour Note that the bottleneck station is neither the one with the slowest machine nor the one with the fewest machine The complete processing time of the line is still the sum of the processing times for one product Thus T0 20 hours Regardless of whether the line has single or multiplemachine stations the critical WIP level is always defined the same way thus W0 rb b 0420 8 units Again since there is still no variability in the line the congestion coefficient for this factory is also OL 0 Best Case Performance for Factory 1 To understand the behavior of Factory l we will simulate several scenarios The first scenario is the case where we allow only one job into the line at a time The first job enters the line at time zero and eXits the line 8 hours later at which time the second job enters the line etc Obviously the throughput of this line under this scenario is 18 jobs per hour Also note that the cycle time is still T 0 8 hours and the throughput is one fourth of the bottleneck rate rb 05 jobs per hour The second scenario is as follows We introduce two successive jobs into the line immediately after the first machine completes the processing of the first job it start working on the second job This process of second job following the first job will continue down to the last operation Except the initial waiting of the second job in front of the first machine while it is processing the first job the second job never waits for processing ever again Moreover since 2 jobs eXit the line in every 8 hours the throughput increased to 28 units per hour Double that when WIP level was one and 50 of the line capacity 173 05 Now in the third scenario we start with three jobs immediately available in front of station 1 and they follow each other as in the second scenario The cycle time is still 8 hours but throughput increased to 38 units per hour or 75 of rb Finally in the fourth scenario we add the fourth job in front of machine 1 and repeat the process We now see that all the stations stay busy all the time once the system is in steady state Because there is no waiting at the stations the cycle time is still 8 hours and it produces one penny in every two hours thus the throughput is now 12 units per hour which equals the line capacity 17 This very special behavior in which the cycle time is T 0 its minimum value and throughput is rb its maXimum value is only achieved when the WIP level is set at the critical WIP level which for Factoryl was evaluated as W0 rbT0 058 4 units The fifth scenario is the case where we now add the fifth job in front of the line and process 5 units at a time Because there are only four machines one job will always wait in front of the first machine Since the measure of cycle time is the time between a job is introduced to the line until the time it leaves the line it now becomes 10 hours due to an extra 2 hour wait in front of machine 1 Thus for the first time now the cycle time is larger than the minimum cycle time of 8 hours However since all the stations are busy the throughput remains at rb 05 At this point the conclusion is that each job we add increases the cycle time by two hours with no increase in throughput The following table summarizes 10 different scenarios and their associated performance measures The graphical presentation of the table results are given in the two figures below LITTLE39S LAW The Little39s law provides the fundamental relationship between WIP cycle time and throughput Law 1 Little39s law TH WIPCT It turns out that this law is valid for all production lines not just those with zero variability Moreover it also applies to a single machine a line or the entire factory Some important uses of the Little39s law follow 1 Queue length calculations Since the law applies to individual stations then we can calculate the eXpected queue length at each station as well as station utilization For example in Factory 2 suppose it was running at the bottleneck rate of 04 jobs per hour From Lttle39s Law we calculate the WIP to be for station 1 04 jobshourquotlt X 2 hours 08 jobs Since there is one machine in station 1 then it is utilized by 80 of the time Similarly at station 3 the Little39s Law predicts an average WIP of 4 jobs Since there are 6 machines at this station the average machine utilization will be 46 X 100 667 Note that the same ratio may be obtained by the ratio of the rate of bottleneck to the rate of station 0406 2 Cycle time reduction From CT WIPT H we deduce that reducing cycle time implies reducing WIP provided that the throughput remains the same Hence large queues are a sign of opportunity to improve cycle time as well as WIP 3 Measure of Cycle Time It is difficult to measure the cycle time since we must keep track of each unit in the system Instead throughput and WIP is tracked routinely Thus we can then use the Little39s Law to estimate CT given the other two quantities 4 Planned Inventory Particularly as a just in time supplier we may have to maintain some finished good inventories to supply our customers on time This may be eXpressed as the number of days of sales inventory storage If 11 days worth of finished goods are to be stocked then the amount of finished goods inventory FGI is given by FGI n X TH Law 2 Best case performance The minimum cycle time CTbest for a given WIP level w is given by T0 ifw 3 W0 CTbest i L wrb otherwise The maximum throughput THbest for a given WIP level w is given by lwTo ifw 3 W0 THbest l rb otherwise Worst Case Performance Note that in the previous analysis we assumed that one work started immediately after one is completed One way to insure that is to carry more than one job to each station on a pallet Whenever a job is finished it is removed from its pallet and the pallet returns back to the beginning of the line to carry the next batch The WIP level is therefore equal to the number of pallets Consider now a real scenario Four pallets are recycled through the system with one additional information Suppose that instead of each job requiring exactly two hours at each station only the job on pallet 1 requires 8 hours at each station while pallets 2 3 and 4 require zero hours The average processing time at each station is 8 0 0 04 2 hours as before and we still have rb 05 jobshour and T0 8 hours However every time pallet 4 reaches a station it finds itself behind pallets l 2 and 3 This is absolute maXimum amount of wait each pallet can have and thus represents the worst case The cycle time for this system is 888832hours or 4T0 and since 4 jobs are output each time pallet l finishes on station 4 the throughput is 432 18 jobs per hour or lTo jobs per hour Also notice that the product of cycle time and throughput is 32 X l 8 4 which is the WIP level so as always the Little39s Law holds The conclusion is therefore Law 3 Worst Case Performance The worst case cycle time for a given WIP level w is given by CTworst WTO The worst case throughput for a given WIP level w is given by THworst 1T0 Practical Worst Case Performance No system ever performs under best case or the worst case scenarios In order to have a practical worst case we will introduce some randomness to the system First we need a definition for quotsystem statequot Simply put a state is a description of location of jobs at stations We will provide the possible states of our system with four machines and three jobs in the table below state vector vector 0 0 l 0 l 2 3 4 5 6 7 8 9 l Depending on the specific assumptions about the line not all states will necessarily occur If all processing times in the four station three job system are one hour and it behaves according to the best case then only four states lll0 01ll l0ll and ll0l will be repeated over and over Similarly if it behaves according to the worst case then four different states 3000 0300 0030 and 0003 will be repeated over and over When randomness is introduced into a line more states become possible suppose that the processing times are still deterministic but every once in a while a machine breaks down for several hours Then some of those states which were not possible in best or worst cases will start occurring As the breakdown becomes more and more prevalent then system will almost surely be in all of those states at some point in time Thus we define the maximum randomness scenario to be that which causes every possible state to occur with equal frequency To achieve this scenario we need the following assumptions about the line 1 The line must be balanced 2 All stations must consist of single machines 3 Process times must be random with eXponential distribution To understand how the PWC works suppose there are N single machine stations each with an average processing time of t and a constant level w of jobs in the line Thus the total processing time for one unit of this line is To Nt and the bottleneck rate is rb lt Since each state is equally likely each time a pallet arrives at a station it will see on average the wl other jobs distributed among the N stations so the eXpected number of jobs ahead of you upon arrival is wlN How much time will a job spend at a station is calculated by waiting for the jobs ahead of it on that station plus its own processing time or Average time at a station wlNt t l wlNt In this formula we have invoked the memoryless property of the eXponential distribution That is we ignored the fact that the job in progress was partially finished when the job under consideration arrived at a station Finally since stations are assumed to be identical the average cycle time is simply the average time at each station times the number of stations CT N1wlNt Nt wlt TO wlrb To get the corresponding throughput applying Little39s law we get TH WIPCT wTo wlrb WWol b Wl1 b wWo W 1rb Definition Practical worst case performance The practical worst case PWC cycle time for a given WIP level w is given by Cprc TO wlrb The PWC throughput for a given WIP level w is given by Tprc wWo w lrb Once we know the best worst and practical worst case performance of a production line we can then compare the actual performance of our system against the limits above If the actual performance is between the best and the practical worst then it is a fairly good line If on the other hand the actual performance lies between the PWC and the worst case then it is a relatively bad line and improvements are possible To improve a bad line we look at the three assumptions we used 1 Balanced lines 2 Single machine stations 3 Exponential processing times Since these three assumptions provided the maximum randomness in the system improving any of them will tend to improve the performance of the line To unbalance a line we can provide additional capacity at a station There may be several feasible ways to accomplish this These options may range from adding a new equipment to reducing downtime to improving the process to redesigning the product to change the processing time andor its distribution and variance to name a few These issues of improving line performance will be discussed in detail at a later time Measuring Congestion We will first state that the congestion in a production line is a function of its WIP level w indicating the average cycle time that is CT cw We can then use the ratio of the actual cycle time at the critical WIP level cWo to the best possible cycle time T0 as an indicator of congestion ie OL cwT0 The logic is that a congested line will have a long cycle time and hence a large ratio The least congested system the best case will have cWo T0 so the ratio is 1 To force oc to a range between 0 and l we normalize this measure We use the measure cwT0 1 instead Second to make OL l for PWC notice that CWo T0 W0 1rb for the PWC so that for this case CW0T0 1 W0 1W0 Therefore if we further adjust the modified ratio by multiplying by WoWo 1 it will equal 1 for PWC We therefore define oc to be 06 WoWo 1CWT0 1 Now since OL 0 for the best case and OL l for the practical worst case since cW0 WoTo by the Worst Case Law it turns out that OL W0 for the worst case Now with this modification if 0c is between 0 and 1 then the line is a fairly good line If on the other hand 0c is above 1 then the line is on the quotbadquot side and therefore a candidate for improvement To see this consider Factory 2 Suppose all processes have zero variability Then it is easy to simulate the line to find that the WIP is equal to the critical WIP W0 8 jobs The cycle time remains at T0 20 hours and so the congestion coefficient is OL 0 Next suppose that all processing times are eXponentially distributed In order to find the cycle time with WIP set at W0 we must simulate the line After performing such a simulation we get an average cycle time of 2579 minutes implying that OL wowo 1 cW0T0 1 Ms 1 257920 1 033 This is substantially less than 1 indicating that the system performs significantly better than the PWC The reason as noted before is the imbalance and the parallel machine stations The net effect is to reduce cycle time by excess capacity The following figures illustrate cycle time and throughput as a function of WIP level for Factory 2 Bottleneck Rates and the Cycle Time The bottleneck rate rb plays an important role in improving production lines since it establishes the capacity of the line First of all if we are operating a good line 0c value less than 1 then the cycle time will be very close to wrb where W is the WIP level Hence increasing the bottleneck rate rb will reduce cycle time for any given WIP level Also note that if we keep increasing the bottleneck rate eventually the bottleneck station will no longer be the bottleneck station but another station will dictate the line39s productivity If we continue improving this station the bottleneck will eventually move to another station This is called the quotshifting bottlenecksquot phenomenon The ultimate destination of this type of journey is to reach to a line which is balanced But from our earlier analysis we know that balanced lines are not the best lines in performance since they have higher 0c value and more congestion than an unbalanced line Finally note that reducing total processing time T0 has little effect on average cycle time for other than extremely low WIP levels ie because cycle times are close to wrb which does not involve T0 Hence by simply speeding nonbottleneck stations will not have much effect in typical plants However when variability is in effect it becomes possible to improve the line performance by improving nonbottleneck stations These issues will be dealt with in future chapters VARIABILTY The most prevalent sources of variability in manufacturing environment are Natural variability which includes minor uctuations in process time due to differences in operators machines and material Random outages Setups Operator availability Recycle


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