Work-sharing in production systems is a modern approach that improves throughput rate. Work is shifted between cross-trained workers in order to better balance the material now in the system. When a serial system is concerned, a common work-sharing approach is the Bucket-Brigade (BB), by which downstream workers sequentially take over items from adjacent upstream work- ers. When the workers are located from slowest-to-fastest and their speeds are deterministic, it is known that the line does not suffer from blockage or starvation, and achieves the maximal theoretical throughput rate (TR). Very little is known in the literature on stochastic self-balancing systems with work-sharing, and on BB in particular. This paper studies the basic BB model of Bartholdi & Eisenstein (1996) under the assumption of stochastic worker speeds. We identify settings in which conclusions that emerge from deterministic analysis fail to hold when speeds are stochastic, in particular relating to worker order assignment as a function of the problem parameters

Bukchin, Yossi

Hanany, Eran

Khmelnitsky, Eugene

English

Georgia Southern University: Digital Commons@Georgia Southern

Georgia Southern UniversityDigital Commons@Georgia Southern14th IMHRC Proceedings (Karlsruhe, Germany –2016) Progress in Material Handling Research2016Throughput Rate of a Two-worker StochasticBucket BrigadeYossi BukchinTel-Aviv University, bukchin@tau.ac.ilEran HananyTel Aviv University, hananye@post.tau.ac.ilEugene KhmelnitskyTel Aviv University, xmel@eng.tau.ac.ilFollow this and additional works at: https://digitalcommons.georgiasouthern.edu/pmhr_2016Part of the Industrial Engineering Commons, Operational Research Commons, and theOperations and Supply Chain Management CommonsThis research paper is brought to you for free and open access by the Progress in Material Handling Research at Digital Commons@Georgia Southern.It has been accepted for inclusion in 14th IMHRC Proceedings (Karlsruhe, Germany – 2016) by an authorized administrator of DigitalCommons@Georgia Southern. For more information, please contact digitalcommons@georgiasouthern.edu.Recommended CitationBukchin, Yossi; Hanany, Eran; and Khmelnitsky, Eugene, "Throughput Rate of a Two-worker Stochastic Bucket Brigade" (2016). 14thIMHRC Proceedings (Karlsruhe, Germany – 2016). 7.https://digitalcommons.georgiasouthern.edu/pmhr_2016/7THROUGHPUT RATE OF A TWO-WORKER STOCHASTICBUCKET BRIGADEYossi BukchinEran HananyEugene KhmelnitskyTel-Aviv UniversityAbstractWork-sharing in production systems is a modern approach that improvesthroughput rate. Work is shifted between cross-trained workers in order tobetter balance the material ow in the system. When a serial system is con-cerned, a common work-sharing approach is the Bucket-Brigade (BB), by whichdownstream workers sequentially take over items from adjacent upstream work-ers. When the workers are located from slowest-to-fastest and their speeds aredeterministic, it is known that the line does not suer from blockage or starva-tion, and achieves the maximal theoretical throughput rate (TR). Very little isknown in the literature on stochastic self-balancing systems with work-sharing,and on BB in particular. This paper studies the basic BB model of Bartholdi& Eisenstein (1996) under the assumption of stochastic worker speeds. Weidentify settings in which conclusions that emerge from deterministic analysisfail to hold when speeds are stochastic, in particular relating to worker orderassignment as a function of the problem parameters.1 IntroductionMass production environments have gone through major changes due to the imple-mentation of work-sharing. Traditionally, such systems were based on division ofwork among multiple workers, each trained to repeatedly perform a small segment ofthe work. Education and training improvements have provided incentives for work-ers to handle more complex requirements, thus providing the potential for enhancing1system performance. Modern environments have applied work-sharing, whereby mul-tiple cross-trained workers are able to perform the same task. The implementationof work-sharing has improved the ability to achieve balanced lines [1, 2], leading tofewer blockages and starvation in the system, and as a result, to a higher throughputrate (TR). Ostolaza et al. [3] were the rst to coin the term Dynamic Line Balancing(DLB). This term refers to the operational side of work-sharing as \allowing tasksto be assigned on the y based on the current state of the system". According toDLB, some of the tasks, dened as `shared tasks' can be shifted between adjacentstations/workers, in order to better balance the ow along the line. Some rules wereconsequently applied to control the line dynamics.One of the most common work-sharing approach is the Bucket-Brigade (BB),proposed by Bartholdi and Eisenstein [4]. This approach initially assumed full cross-training of the workers, and suggested conditions ensuring a self-balancing line. Inthe basic model, they also assumed that a task can be handed from one worker toanother at any point, and that the system is deterministic in all respects. The basicrule of BB is that whenever the last downstream worker completes an item, the workerreturns and takes over the item of the next upstream worker; the preempted workercontinues similarly with the next upstream worker, where this procedure continuesuntil the rst worker is preempted and starts processing a new item. This procedureprevents starvation, however the authors show that when the workers are locatedfrom slowest-to-fastest, the line does not suer from blockage as well, thus achievingthe maximal theoretical throughput rate. Moreover, they show that in steady statethe `work taking over' between any two adjacent workers always occurs at the samepoint, meaning that each worker always executes exactly the same work segment oneach item. Further analysis of the BB system dynamics for two and three workers waspresented in [5], while the chaotic behavior of the hand-o point when the convergencecondition does not hold was studied in [6]. For multiple extensions of BB, see [7].Typical serial systems in which BB can be applied are production/assembly linesand order picking systems. In the latter, multiple pickers, located along an aisle,perform picking tasks from a ow rack (see Figure 1.1). The items of each order arepicked into a box/tote by a picker, while moving toward the end of the aisle. Whenthe last picker completes an order, she moves backward and takes over the box of theadjacent upstream picker, who does the same to the next upstream picker and so on.2Figure 1.1: An order picking system (from Bartholdi et al. 2001)Bartholdi et al. [8] relaxed the deterministic work content assumption, and ex-amined the eectiveness of BB assuming stochastic work times under exponentialdistribution. They show analytically that, as the number of stations increases, themodel converges to the optimal throughput when workers are assigned from slowest-to-fastest. Bratcu and Dolgui [9] studied stochastic speeds via simulations, whileassuming that both working and walk back speeds are normally distributed. Hong[10] suggested a closed-form expression for the 2-worker blocking congestion in acircular-passage system, for constant workers' speed and stochastic pattern of picks.He showed via simulation that the analytical expression provides a good approxima-tion for the blockage probability of a BB picking system, when the worker speed andhand-o time are constant and the picking pattern is stochastic.As opposed to the knowledge on stochastic self-balancing systems without work-sharing, very little is known in the literature on such systems with work-sharing, andon BB in particular. In this paper we analyze the basic model of BB [4] under theassumption of stochastic worker speeds. We present the dynamics of the system,and provide analytic results to the case where one worker dominates the other. Inthe general case, when partial blockage may occur, we demonstrate the eect of theworkers speed parameters on the TR, and show that in some cases, the fastest-to-slowest assignment in terms of expected speeds provides higher TR than the reverse3order.2 ModelBucket Brigade (BB) is a decentralized dynamic protocol, which allows coordinatingthe eorts of several workers along a production line. Each worker moves down theline with an item. Passing downstream workers is not allowed, so each worker eitherproceeds at their own pace or at a reduced speed when blocked by a downstreamworker. The last worker, upon completion of an item at the end of the line, returnsto take over (hand-o) the item from the immediate upstream worker, who does thesame, until the rst worker takes a new item from the start of the line. We assumethat the time required to return upstream is insignicant compared with the timerequired to work downstream.2.1 Bucket Brigade dynamicsThe dynamics of a two-worker system can be described with discrete events, wherehand-o n = 0; 1; 2; :::, refers to the event by which the second worker has reachedthe end of the line completing an item, and returns to receive the item from the rstworker. A state of the system, xn, denes the position along the line of hand-o n, i.e.the position of worker 1 when worker 2 has reached the end of the line. Without lossof generality, we assume that the length of the line is 1, so that 0  xn  1. At eachhand-o, the worker speeds, vi, i = 1; 2, are randomly and independently generatedaccording to known, stationary cumulative distribution functions (cdf) Fi(),  2[0;1).In case worker 1 is blocked by worker 2, the two workers reach the end of the lineconcurrently, and the hand-o takes place at the position xn = 1. Following the BBrule, the next hand-o takes place immediately and with certainty at the positionxn = 0. Since worker 2 cannot be blocked, and always reaches the end of the line,the time elapsed between hand-o n and hand-o n + 1 is always 1 xnv2. Thus, theposition of hand-o n+ 1 is calculated recursively by,xn+1 = min1;v1v2(1  xn); where x0 is given: (2.1)4Analyzing this stochastic system as a Markov chain, we are interested in determiningthe steady state distribution of hand-o positions along the line, the expected cycletime, and the throughput rate. In steady state, the hand-o position, x, satisesx = min1;v1v2(1  x); (2.2)which is obtained from (2.1) by omitting the hand-o index.2.2 Throughput rateAs discussed above, the time elapsed between consecutive hand-os in a BB produc-tion line is 1 xv2, thus the expected cycle time of the line isE [CT ] = E1  xv2= (1  E[x])E1v2; (2.3)where the second equality follows because v2 is drawn independently from the hand-o position, x. The TR of the BB line when the workers are assigned in the order1! 2 , is denoted by TR1!2, and is equal to the inverse of the cycle time,TR1!2 = (1  E[x]) 1E1v2 1: (2.4)From 2.3 we can see that the TR of a single worker i, is equal toTRi = E1vi 1: (2.5)The TR of a BB line, as shown in (2.4), depends on TR2, i.e., Eh1v2i 1, and onthe expected hand-o position, E[x]. Thus TR1!2 combines the eorts of the twoworkers. Dene TR1!2i for i = 1; 2 as the TR of worker i when working in a BB lineconsisting of workers 1 and 2 in this order. The next proposition shows that the TRof worker 1 in a BB line is bounded from above by this worker's expected speed, andthe TR of worker 2 equals the corresponding single worker TR, i.e., does not dependon the distributions of v1 and x.5Proposition 2.1. In a BB line,TR1!21  E[v1]andTR1!22 = TR2:Proof. All proofs are given in [11]Contrary to worker 2, the TR of worker 1 depends on all parameters of the BBline. It is obtained by subtracting TR1!22 from the TR of the line,TR1!21 = TR1!2   TR1!22 = E1v2 111  E[x]   E1v2 1= E1v2 1E[x]1  E[x] :3 Throughput rate in the case of dominanceIn an instance of a BB line with stochastic speeds, worker 1 may be blocked by worker2 in some iterations, resulting in subsequent hand-os at xn = 1 and xn+1 = 0. Inother iterations, where worker 1 is not blocked, the hand-os occur in some 0 < xn <1. This section deals with particular cases of no blockage, where worker 1 is neverblocked, and full blockage, where worker 1 is blocked at each hand-o. These casescome up when one of the workers in the team is almost surely faster than the other,i.e. with probability 1 has a higher or equal speed at each hand-o.Assume, without loss of generality, that worker 2 dominates worker 1, i.e., v1  v2for sure. The two assignments, 1! 2 and 2! 1, correspond to the slowest-to-fastestand fastest-to-slowest rules, respectively, which are well-dened only in the case ofdominance. Denote TRi!j as the throughput rate of a 2-worker BB line with aorkersi and j in that order, and TRi!jk , as the throughput rate of worker k = i; j in thisline. The slowest-to-fastest personnel assignment rule is known to be optimal in termsof TR when workers have deterministic speeds [4]. The rule allows avoiding blockageand attains the maximum TR,TR1!2 = v1 + v2: (3.1)6The opposite, fastest-to-slowest assignment leads to full blockage in the deterministicenvironment, with the TR, TR2!1 = 2v1, slower than that in (3.1).Under stochastic speeds, the question of setting an appropriate personnel assign-ment rule is signicantly more complicated. However, in the case of dominance, theslowest-to-fastest assignment is better than the opposite one, as proved in the nextproposition. The proposition also explicitly calculates the TR of the workers and ofthe line.Proposition 3.1. If worker 2 dominates worker 1, then the slowest-to-fastest assignment yieldsTR1!21 = E [v1]  TR1!22 ; TR1!2 = E [v1] + E1v2 1; (3.2) the fastest-to-slowest assignment yieldsTR2!12 = TR2!11 = E1v1 1; TR2!1 = 2E1v1 1; (3.3) the slowest-to-fastest assignment is better: TR1!2  TR2!1.Proposition 3.1 shows that under dominance, the TR of the rst worker can neverexceed that of the second worker. Additionally, while the TR in BB depends on thewhole distribution of the speed of worker 2, it depends on the speed of worker 1 onlythrough the expected value.4 General Case: Numerical ApproximationIn this section we analyze numerically (using Wolfram Mathematica) an approximatesolution to cases where partial blockage may exist, and gain some additional insightson the stochastic behavior of BB. The speeds of the two workers are taken fromBeta distribution. As an illustration, Figure 4.1 depicts the TR as a function of theexpected worker speeds ei and their standard deviations si. This gure demonstrateshow the conclusions of Proposition 3.1 extend to the case of partial blockage, showingthat the TR increases with both workers' expected speed and decreases with their7standard deviation. The eect of both parameters on the TR is signicantly moresubstantial for the second worker as compared to the rst worker. In particular, theTR is close to zero when the coecient of variation of the second worker's speed isclose to 1. Since the standard deviation of the second worker has signicant eecton the TR of the line, the slowest-to-fastest order based on the expected speeds isnot always optimal. In particular, if both workers have the same expected speed,it is better to located the one with the larger standard deviation rst. However,when there is a relatively large dierence between the standard deviation of the twoworkers, it is often optimal to locate the worker with the highest expected speed rst,namely, fastest-to-slowest order based on the expected speed.Figure 4.1: Throughput rate (e2 = 5 , s2 = 1)References[1] Anuar R. and Bukchin Y., Design and operation of dynamic assembly lines usingwork-sharing, International Journal of Production Research, 44 (18-19), 2006,4043-4065.[2] Bukchin, Y. and Sofer, T., Bi{Directional Work-Sharing in Assembly Lines withStrict and Flexible Assembly Sequences, International Journal of Production8Research 49(14), 2011, 4377-4395.[3] Ostolaza J., McClain J.O. and Thomas J., The use of dynamic (state-dependent)assembly line balancing to improve throughput. Journal of Manufacturing Op-erations Management 3, 1990, 105{133.[4] Bartholdi, J.J. and Eisenstein, D.D., A production line that balances itself. Op-erations Research 44 (1), 1996, 21{34.[5] Bartholdi J.J, Bunimovich, N.A. and Eisenstein D.D., Dynamics of two- andthree-worker \bucket brigade" production lines, Operations Research 47(3),1999, 488-491.[6] Bartholdi, J.J., Eisenstein, D.D. and Lim, Y.F., Deterministic chaos in a modelof discrete manufacturing, Naval Research Logistics 56(4), 2009, 293-299.[7] Bratcu, A.I. and Dolgui, A., A survey of the self-balancing production lines(\bucket brigades"), Journal of Intelligent Manufacturing 16, 2005, 139{158.[8] Bartholdi J.J, Eisenstein D.D. and Foley R.D, Performance of bucket brigadeswhen work is stochastic, Operations Research 49(5), 2001, 710{719.[9] Bratcu, A.I. and Dolgui, A., Some new results on the analysis and simulationof bucket brigades (self-balancing production lines), International Journal ofProduction Research 47(2), 2009, 369-387.[10] Hong S., Two-worker blocking congestion model with walk speed m in a no-passing circular passage system. European Journal of Operational Research235, 2014, 687{696.[11] Bukchin, Y, Hanany, E. and Khmelnitsky, E., Bucket Brigade with StochasticWorker Pace, working paper, Dept. of Industrial Engineering, Tel-Aviv Univer-sity. 2016.9

Throughput Rate of a Two-worker Stochastic Bucket Brigade

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Throughput Rate of a Two-worker Stochastic Bucket Brigade

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