FIFTH INTERNATIONAL CONFERENCE ON“ANALYSIS OF MANUFACTURING SYSTEMS-PRODUCTION MANAGEMENT”

MAY 20-25 2005, ZAKYNTHOS ISLAND, GREECE

“On the optimization of Server allocation in large reliable production lines with exponential processing times”

A. C. Diamantidis (adiama@syros.aegean.gr)UNIVERSITY OF THE AEGEAN

DEPARTMENT OF PRODUCT AND SYSTEMS DESIGN ENGINEERING SYROS ISLAND, GREECE

C. T. Papadopoulos (hpap@econ.auth.gr)ARISTOTLE UNIVERSITY OF THESSALONIKI

DEPARTMENT OF ECONOMICSTHESSALONIKI, GREECE

2

PRPRESENTATION OUTLINEESENTATION OUTLINE • PROBLEM DEFINITION

• ASSUMPTIONS OF THE MODEL

• THE PERFORMANCE EVALUATION TECHNIQUE

• THE OPTIMIZATION TECHNIQUE

• NUMERICAL RESULTS FOR THE SERVER ALLOCATION PROBLEM

• FINDINGS

• FURTHER RESEARCH

• REFERENCES

3

PROBLEM DEFINITIONPROBLEM DEFINITION

• This study examines a constrained optimisation problem in designing large production lines with reliable multiple identical parallel machine workstations.

• The problem is how to allocate a total number of servers among all workstations in order to maximize the throughput of the production line. This problem is known as the sever allocation problem (SAP).

4

A production line with reliable multiple identical parallel machine workstations

Figure 1: A flow line with N parallel-machine workstations and N-1 intermediate buffers. 

5

ASSUMPTIONS OF THE MODELASSUMPTIONS OF THE MODEL

1. Each workstation Mi in figure 1, consists of Si reliable and identical parallel machines. At each workstation, each one of the Si parallel machines has exponentially distributed service time with mean 1/μi, i=1,…,N.

2. The parallel machines of different work stations are not necessarily identical, viz., station processing times are assumed to be exponentially distributed with non identical mean service rates.

3. It is also assumed that when any one of the Si parallel machines at workstation Mi completes a part, that part is placed in the buffer Bi downstream of the workstation immediately, provided the buffer is not full.

6

LITERATURE REVIEWLITERATURE REVIEW• There is a relatively scarce literature concerning the

buffer allocation and server allocation problem for flow lines with multiple parallel-machine workstations.

• Magazine and Stecke (1996) considered small flow lines with two and three workstations consisting of parallel machines. They examined how the throughput of such systems may be improved if specific parameters of the system such as the allocation of machines among the workstations, allocation of workload to the workstations and buffer allocation between workstations vary. Considering the (SAP) for lines with three workstation they found that the middle station needs to be more efficient, so it takes the largest number of the available servers.

7

• Hillier and So (1989) examined how the assignment of extra servers to small (up to eight workstations) production lines with small or no buffers maximizes the throughput of such systems. With a fixed number of extra servers over an equal allocation to all stations they focused on the question of where to place these extra servers in order to maximize throughput . The main conclusion was that the interior stations (especially the center stations) should be given priority over the end stations for allocating an extra server. They also presented numerical results for Erlang, exponential and two-stage Coxian service time distributions.

• Hillier and So (1995) considered tandem queueing systems that could be formulated as a continuous time Markov-chain. They considered various constrained optimization problems such as buffer allocation problem (BAP), SAP and the workload allocation problem (WAP). Each decision variable was examined either in isolation or simultaneously with each one of the other two variables. They considered production lines where the maximum number of workstations is up to eight and the buffer capacities between the workstations are one or zero. For the SAP their main finding was that the extra servers (above a uniform allocation) should go to the centre workstations rather than the first or last workstation.

8

• Hillier and So (1996) studied the problem of the simultaneous optimization of server and work allocation of small serial production lines. The most important finding of their study is the L-phenomenon according to which every station receives just a single server except for one of the two end stations which receives all the extra servers.

• The optimal allocation of parallel servers with different non-exponential service time distributions at each workstation was considered by Futamura (2000). The effect of the coefficient of variation (cv) of the service time distribution on the throughput of systems, where cv varies from one workstation to another, was examined

9

EVALUATION AND OPTIMIZATION TECHNIQUESEVALUATION AND OPTIMIZATION TECHNIQUES

PERFORMANCE PERFORMANCE EVALUATIEVALUATION ON OPTIMIZATIONOPTIMIZATION

TECHNIQUES TECHNIQUES TECHNIQUES TECHNIQUES ((GENERATIVEGENERATIVE)

1) MARKOV CHAIN 1) ENUMERATION2) DECOMPOSITION 2) HEURISTIC3) SIMULATION 3) SIMULATED ANNEALING4) PETRI NET 4) GENETIC ALGORITHMS5) OTHER METHODS 5) TABU SEARCH

6) OTHER METHODS

10

PERFORMANCE EVALUATION TECHNIQUE:THE DECOMPOSITION METHOD

• The use of Markovian analysis for the numerical computation of the performance measures of large systems like the one depicted in Figure 1 is almost impossible due to the enormous resulting state space.

• In this study, the classical decomposition method, proposed by Gershwin (1987) and Dallery et al (1988) was applied in order to evaluate the performance measures of production lines like the one depicted in figure 1. The decomposition equations were modified to take into account the existence of parallel machines at each workstation.

11

• Ancelin and Semery (1987) described a method that replaces each parallel-machine workstation by an equivalent single machine workstation. The processing rate of the equivalent workstation equals the sum of the processing rates of all parallel machines in the workstation.

• Burman (1995) presented a method that replaces each parallel server workstation by a single equivalent workstation for the case of continuous flow of material. He assumes that the equivalent workstation has a maximum processing rate which equals the sum of the processing rates of the parallel machines.

• Patchong and Willaeys (2001) presented a technique that replaces each parallel machine workstation by an equivalent single machine workstation for the case of serial production lines. The sets of equations that are necessary for this replacement are derived.

• The difference of the proposed approach and those of Ancelin and Semery (1987), Burman (1995) and Patchong and Willaeys (2001), is that each parallel-machine workstation is not replaced by an equivalent workstation. That is, the decomposition approach is applied directly to each one of the parallel machines for each workstation without using replacement techniques.

12

SOLUTION OF LARGE SYSTEMS WITH PARALLEL MACHINES AT EACH WORKSTATION

• In order to develop the decomposition method, for each buffer Bi between two workstations Mi and Mi+1 with Si and Si+1parallel machines respectively , a virtual upstream workstation Mu(i) that represents the flow of material into this buffer has to be introduced.

• Similarly a virtual downstream workstation Md(i) that represents the flow of material out of buffer has to be introduced.

• Workstation Mu(i) consists of Si virtual machines while workstation Md(i) consists of Si+1virtual downstream machines.

• The service times of the Si parallel machines of pseudo workstation Mu(i) are exponentially distributed with mean 1/μu(i), while the service times of the Si+1 parallel machines of pseudo workstation Md(i) are also exponentially distributed with mean 1/μd(i), i=1,…,N-1.

See figure2

13

THE SETS OF THE DECOMPOSITION EQUATIONS

• Since, in our model, all Si parallel machines of each workstation Mi in the real line L, are reliable, only two sets of decomposition equations are derived. These are: the conservation of flow equations and the flow rate idle time equations.

• CONSERVATION OF FLOW EQUATIONS

The fact that the flow is conserved because there is no mechanism for the creation or destruction of material, leads to the conservation of flow equations.

• FLOW RATE IDLE TIME EQUATIONS

These sets of equations are used to calculate the parameters μu(i) and μd(i-1) , i=2,…,N-1.

See the decomposition equations

14

• An algorithm that simultaneously solves all the derived sets of the decomposition equations in order to evaluate the performance measures of large flow lines with parallel-machine workstations was developed.

• The numerical results indicate that the decomposition algorithm is very accurate. The average percentage error of the throughput obtained from the proposed decomposition algorithm and simulation (in Arena) for lines with up to 100 stations is less than 1.2%, whereas the results for lines with up to 1000 stations indicate that the percentage error is less than 2.5%.

15

SIMULATED ANNEALING

• Simulated annealing is an adaptation of the simulation of physical thermodynamic annealing principles described by Metropolis et al. (1953) to the combinatorial optimization problems (Kirkpatrick et al. 1983, Cerny 1985).

• The simulated annealing method starts with a non-optimal initial configuration (which may be chosen at random) and works on improving it by selecting a new configuration using a suitable mechanism (at random in the simulated annealing case) and calculating the corresponding cost differential (ΔR). If the cost is reduced, then the new configuration is accepted and the process repeats until a termination criterion is satisfied.

16

• Unfortunately, such methods can become `trapped’ in a local optimum that is far from the global optimum. Simulated annealing avoids this problem by allowing`uphill’ moves based on a model of the annealing process in the physical world.

17

NUMERICAL RESULTS FOR THE SERVER ALLOCATION PROBLEM

• For all cases examined in this study the unit of time is normalized by setting the average of the μj, j=1,…,N, values equal to 1.

• All the buffer capacities qi, i=1,…,N-1 are assumed equal to zero. The throughput of the system is denoted PR(s).

• The optimal server allocation among the N workstations is denoted by the vector s= (s1,…,sN).

• The parameters of the numerical experiments are summarized in the following two tables

18

Parameter RangeN even 6,8,..,30(2)

36,40,46,56 and 50,…,100(10)S Varies from 7 to 101qi 0 for all i=1,…,N-1

E 1μj 1 for all j=1,…,N

Parameter Range

N odd 7,9, 15,…,29(2) 35,…,55(10)41,…61(10)

S Varies from 8 to 62

qi 0 for all i=1,…,N-1

E 1μj 1 for all j=1,…,N

19

• Similarly to Hillier and So (1989) we define n and E as follows:n is the smallest integer that is less than or equal to S/N and

E=S-n Ni.e., the feasible values of E are E=0,1,…,N-1.

Quantity E actually represents the number of “extra” servers that are available for allocation beyond a uniform allocation of n servers to each station.

See the numerical results

20

FINDINGS

• The numerical experiments have pointed out that PR(s1,s2,…,sN)PR(sN,sN-1,…,s1).

• Considering the results given in Table 1, for E=1 and N even, the optimal solution assigns the one extra server to one of the two center stations. The only difference between this design rule and the one presented by Hillier and So (1989) is that the optimal solution is not always the one that assigns the extra server to the back center station.

• Considering the results given in Table 2 for E=1 and N odd the throughput is maximized by assigning the one extra server to the center station. This design rule is identical to the one presented in Hillier and So (1989) for small production lines.

21

CONCLUSIONS• In this study the server allocation problem for large

production lines with multiple parallel machine workstations was examined.

• An extension of the original decomposition method that is applicable for large production lines with multiple parallel machine workstations developed by Diamantidis, Papadopoulos and Heavey (2005) was used as evaluative technique, whereas simulated annealing was used as generative technique.

• The design rules presented by Hillier and So (1989) for small lines (up to eight stations) with multiple parallel machine workstations are extended to large (up to 100 stations) production lines of the same type.  

22

FURTHER RESEARCH

• The authors are currently working on the buffer allocation problem and the simultaneous optimization of the server and buffer allocation for large production lines with multiple parallel machine workstations.

23

REFERENCES

[1] Ancelin, B. and Semery, A. (1987), ``Calcul de la productivité d'une ligneintégrée de fabrication: CALIF, une méthode analytique industrielle’’, RAIRO APII, 21(3), 209--238.[2] Burman, Mitchell H. (1995), “New Results in Flow Line Analysis”, Ph.D. Thesis, Operations Research Center, Massachusetts Institute of Technology, USA.

[3] Cerny,V., 1985, Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. Journal of Optimization Theory and Applications, 45, 41-51.[4] Dallery, Y., David, R. and Xie, X. (1988), “An efficient algorithm for the analysis of transfer lines with unreliable machines and finite buffers”, IIE Transactions, 20(3), 280– 283. [5] Futamura, K. (2000), “The multiple server effect: Optimal allocation of servers to stations with different service – time distributions in tandem queuing networks”, Annals of Operations Research, 93, 71–90.[6] Gershwin, S.B. (1987), “An efficient decomposition algorithm for the approximate evaluation of tandem queues with finite storage space and blocking”, Operations Research, 35, 291–305.

24

[7] Hillier, Frederick S. and So, Kut C. (1989), “The Assignment of Extra Servers to Stations in Tandem Queueing Systems with Small or No Buffer”, Performance Evaluation, 10, 219-231.

[8] Hillier, Frederick.S.and So, Kut.C (1995), “On the optimal design of tandem queueing systems with finite buffers”, Queueing Systems 21, 245-266.

[9] Hillier, Frederick S. and So, Kut C. (1996), “On the simultaneous optimization of server and work allocations in production line systems with variable processing times”, Operations Research, Vol. 44, No. 3, 435-443.

[10] Kirkpatrick, S., Jr.,Gelatt,C.D., and Vecchi,M. P., 1983, Optimization by simulated annealing. Science, 220, 671- 679.

[11] Magazine, M.J. and Stecke, K.E. (1996), “Throughput for production lines with serial workstations and parallel service facilities”, Performance Evaluation, 25, 211–232.

[12] Metropolis,N.,Rosenbluth,A.N.,Rosenbluth,M.N.,Teller,A.H., and Teller,E.,1953, Equation of state calculation by fast computing machines. Journal of Chemical Physics, 21, 1087- 1092.

[13] Patchong, A. and Willaeys, D. (2001), “Modelling and analysis of an unreliable flow line composed parallel – machine stages”, IIE Transactions, 33, 559–568.