# Double and Triple Optimization

Chapter

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## Abstract

There are three pure allocation problems, viz., the work-load allocation problem, the server allocation problem and the buffer allocation problem, all concerned with maximizing throughput. Mathematically, these problems may be described as follows:

The work-load allocation problem, WAP: subject to: for normalized total work-load equal to unity and fixed allocation of servers and fixed buffer allocation.

$$\max X({\bf w}) =\max X({w}_{1},{w}_{2}, \ldots ,{w}_{K})$$

$$\sum _{i=1}^{K}{w}_{ i} = 1,\ \ \ \mbox{ for ${w}_{i} > 0$}$$

The server allocation problem, SAP: subject to: for fixed allocation of work to each station and fixed buffer allocation.

$$\max X({\bf s}) =\max X({S}_{1},{S}_{2}, \ldots ,{S}_{K}))$$

$$\sum _{i=1}^{K}{S}_{ i} = S,\ \ \ \mbox{ for ${S}_{i} \geq 1$ and integer}$$

The buffer allocation problem, BAP: subject to: for fixed allocation of work to each station and fixed allocation of servers.

$$\max X({\bf n}) = X({N}_{2}, \ldots ,{N}_{K})$$

$$\sum _{i=2}^{K}{N}_{ i} = N,\ \ \ \mbox{ for ${N}_{i} \geq 0$ and integer}$$

As indicated above, there are three single-variable decision problems. Combining these problems into two-variable problems leads to the following three problems which may be mathematically described as follows:

The combined work-load allocation and server allocation problems, W + S: subject to: and and for fixed buffer allocation.

$$\max X({\bf w},{\bf s})$$

$$\sum _{i=1}^{K}{w}_{ i} = 1,\ \ \ \mbox{ for ${w}_{i} > 0$ and normalized work-load}$$

$$\sum _{i=1}^{K}{S}_{ i} = S,\ \ \ \mbox{ for ${S}_{i} \geq 1$ and integer}$$

The reader may note that this problem has already been discussed in Chapter 4.

## Keywords

Simulated Annealing Allocation Problem Expansion Method Service Time Distribution Complete Enumeration
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

- 1.Buzacott, J.A. and Shanthikumar, J.G. (1993),
*Stochastic Models of Manufacturing Systems*, Prentice Hall.Google Scholar - 2.Dallery, Y. and Frein, Y. (1993), On decomposition methods for tandem queueing networks with blocking,
*Operations Research*, Vol. 41, No. 2, pp. 386–399.CrossRefzbMATHMathSciNetGoogle Scholar - 3.Diamantidis, A.C., Papadopoulos, C.T., and Heavey, C. (2006), Approximate analysis of serial flow lines with multiple parallel-machine stations,
*IIE Transactions*, Vol. 39, issue 4, pp. 361–375.CrossRefGoogle Scholar - 4.Futamura, K. (2000), The multiple server effect: Optimal allocation of servers to stations with different service-time distributions in tandem queueing networks,
*Annals of Operations Research*, Vol. 93, pp. 71–90.CrossRefzbMATHMathSciNetGoogle Scholar - 5.Gershwin, S.B. and Schor, J.E. (2000), Effcient algorithms for buffer space allocation,
*Annals of Operations Research*, Vol. 93, 1/4, pp. 117–144.CrossRefzbMATHMathSciNetGoogle Scholar - 6.Heavey, C., Papadopoulos, H.T., and Browne, J. (1993), The throughput rate of multistation unreliable production lines,
*European Journal of Operational Research*, Vol. 68, pp. 69–89.CrossRefzbMATHGoogle Scholar - 7.Hillier, F.S. and So, K.C. (1989), The assignment of extra servers to stations in tandem queueing systems with small or no buffers,
*Performance Evaluation*, Vol. 10, pp. 219–231.CrossRefGoogle Scholar - 8.Hillier, F.S. and So, K.C. (1995), On the optimal design of tandem queueing systems with finite buffers,
*Queueing Systems*, Vol. 21, pp. 245–266.CrossRefzbMATHMathSciNetGoogle Scholar - 9.Hillier, F.S. and So, K.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, pp. 435–443.CrossRefzbMATHGoogle Scholar - 10.Jain, S. and Smith, J.M. (1994), Open finite queueing networks with \(M/M/C/K\) parallel servers,
*Computers & Operations Research*, Vol. 21, No. 3, pp. 297–317.CrossRefzbMATHGoogle Scholar - 11.Kerbache, L. and MacGregor Smith, J. (1987), The generalized expansion method for open finite queueing networks,
*European Journal of Operational Research*, Vol. 32, pp. 448–461.CrossRefzbMATHMathSciNetGoogle Scholar - 12.Lau, H.-S. and Martin, G.E. (1986), A decision support system for the design of unpaced production lines,
*International Journal of Production Research*, Vol. 24, No. 3, pp. 599–610.CrossRefGoogle Scholar - 13.Magazine, M.J. and Stecke, K.E. (1996), Throughput for production lines with serial work stations and parallel service facilities,
*Performance Evaluation*, Vol. 25, pp. 211–232.CrossRefzbMATHGoogle Scholar - 14.Papadopoulos, C.T. and Karagiannis, T.I. (2001), A genetic algorithm approach for the buffer allocation problem in unreliable production lines,
*International Journal of Operations and Quantitative Management*, Vol. 7, No. 1, pp. 23–35.Google Scholar - 15.Spinellis, D.D. and Papadopoulos, C.T. (2000a), A simulated annealing approach for buffer allocation in reliable production lines,
*Annals of Operations Research*, Vol. 93, pp. 373–384.CrossRefzbMATHMathSciNetGoogle Scholar - 16.Spinellis, D.D. and Papadopoulos, C.T. (2000b), Stochastic algorithms for buffer allocation in reliable production lines,
*Mathematical Problems in Engineering*, Vol. 5, issue 6, pp. 441–458.CrossRefzbMATHMathSciNetGoogle Scholar - 17.Spinellis, D., Papadopoulos, C., and MacGregor Smith, J. (2000), Large production line optimization using simulated annealing,
*International Journal of Production Research*, Vol. 38, No. 3, pp. 509–541.CrossRefzbMATHGoogle Scholar - 18.Tempelmeier, H. (2003), Simultaneous buffer and work-load optimization for asynchronous flow production systems, in
*Proceedings of the Fourth Aegean International Conference on the Analysis of Manufacturing Systems*, July, 1–4, 2003, Samos Island, Greece, pp. 31–39.Google Scholar

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