Abstract
In this chapter models applicable during the basic planning stage for the optimal design of flexible manufacturing systems are studied. An overview of the models already existing in literature is presented together with new models developed by the author.
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Literature
Secco-Suardo, G.: Optimization of closed queueing networks, in: Complex Materials Handling and Assembly Systems Final Report Vol. III No. ESL-FR-834–3, Electr. Syst. Lab. M.I.T., Cambridge MA, July 1978, pp.31–32;
Secco-Suardo, G.: Workload Optimization in a FMS Modelled as a Closed Network of Queues, in: Annals of the CIRP, 28(1979)1, p.382
Kimemia, J.G., Gershwin, S.B.: Multicommodity Network Flow Optimization in Flexible Manufacturing Systems, in: Complex Materials Handling and Assembly Systems Final Report Vol. II No. ESL-FR-834–2, Electr. Syst. Lab. M.I.T., Cambridge MA, July 1978, pp.35–37;
Kimemia, J.G., Gershwin, S.B.: Network Flow Optimization in Flexible Manufacturing Systems, in: Proc. IEEE Conf. on Decision and Control 1979 pp.633–639;
Kimemia, J.G., Gershwin, S.B.: Flow Optimization in Flexible Manufacturing Systems, in: IJPR 23(1985)1, pp.81–96
Avonts, L.H., Gelders, L.F., Wassenhove Van, L.N.: Allocation work between an FMS and a conventional jobshop: A case study, in: EJOR, 33(1988), pp.245–256
see chapter 4.2.2
The model presented is an essence out of two publications on the model: Kimemia, J.G., Gershwin, S.B.: Multicommodity Network Flow Optimization in Flexible Manufacturing Systems, in: Complex Materials Handline and Assembly Systems Final Report Vol. II No. ESL-FR-834–2, Electr. Syst. Lab. M.I.T., Cambridge MA, July 1978, pp.31–32;
Kimemia, J.G., Gershwin, S.B.: Network Flow Optimization in Flexible Manufacturing Systems, in: Proc. IEEE Conf. on Decision and Control 1979 pp.634–635; Note that in the second paper the fixation of the production rate by constraint sets (2.2) and (2.4) has to be eliminated if our aim is to maximize total production in the objective function (2.10). It is only necessary to fix the relative amounts of production between different parts (see constraint (2.31) or (2.28) in the first paper). Further the continuity of operational flow must be ensured and is missing in the second paper (see constraint set (2.27) in the first paper).
Kimemia, J.G., Gershwin, S.B.: Multicommodity Network Flow Optimization in Flexible Manufacturing Systems, in: Complex Materials Handling and Assembly Systems Final Report Vol. II No. ESL-FR-834–2, Electr. Syst. Lab. M.I.T., Cambridge MA, July 1978, p.25
Secco-Suardo, G.: Optimization of closed queueing networks, in: Complex Materials Handling and Assembly Systems Final Report Vol. III No. ESL-FR-834–3, Electr. Syst. Lab. M.I.T., Cambridge MA, July 1978, pp.21–22;
Gerla, M.: The design of store-and-forward (S/F) networks for computer communications, Ph.D. thesis Dep. Comp. Science, Univ. Calif. Los Angeles, 1973;
Cantor, D.G., Gerla, M.: Optimal Routing in a Packet Switched Computer Network, in: IEEE Trans, on Computers, C-23(1974)10, pp.1062–1069;
Kimemia, J.G., Gershwin, S.B.: Multicommodity Network Flow Optimization in Flexible Manufacturing Systems, in: Complex Materials Handling and Assembly Systems Final Report Vol. II No. ESL-FR-834–2, Electr. Syst. Lab. M.I.T., Cambridge MA, July 1978, pp.47,
Kimemia, J.G., Gershwin, S.B.: Multicommodity Network Flow Optimization in Flexible Manufacturing Systems, in: Complex Materials Handling and Assembly Systems Final Report Vol. II No. ESL-FR-834–2, Electr. Syst. Lab. M.I.T., Cambridge MA, July 1978, pp. 51–63;
Kimemia, J.G., Gershwin, S.B.: Network Flow Optimization in Flexible Manufacturing Systems, in: Proc. IEEE Conf. on Decision and Control 1979 pp.635–636;
Kimemia, J.G., Gershwin, S.B.: Multicommodity Network Flow Optimization in Flexible Manufacturing Systems, in: Complex Materials Handling and Assembly Systems Final Report Vol. II No. ESL-FR-834–2, Electr. Syst. Lab. M.I.T., Cambridge MA, July 1978, p.25
Yao, D.D., Shanthikumar, J.G.: The optimal input rates to a system of manufacturing cells, in: INFOR, 25(1987)1, pp.57–65
Yao, D.D., Shanthikumar, J.G.: Some resource allocation problems in multi-cell systems, in: Proc. 2nd ORSA/TIMS Conf. on Flexible Manufacturing Systems: Operations Research Models and Applications, edited by ICE. Stecke and R. Suri, Amsterdam 1986, pp.250–253
Yao, D.D., Shanthikumar, J.G.: Some resource allocation problems in multi-cell systems, in: Proc. 2nd ORSA/TIMS Conf. on Flexible Manufacturing Systems: Operations Research Models and Applications, edited by K.IE. Stecke and R. Suri, Amsterdam 1986, p.250
for the proof of convexity of the model and a proof for convergence of the algorithm to the optimal solution see: Yao, D.D., Shanthikumar, J.G.: The optimal input rates to a system of manufacturing cells, in: INFOR, 25(1987)1, pp.57–65;
Kobayashi, H., Gerla, M.: Optimal Routing in Closed Queueing Networks, in: ACM Trans. Comp. Syst., 1(1983)4, pp.294–310; For a description of the FD-Algorithms, see chapter 6.1.1.2.
Little, J.D.C.: A Proof of the Queueing Formula L = λ • W, in: OR 9(1961), pp.383–387
See definitions of workloads in chapter 6.1.2.2.2
Stecke, K.E.: On the Nonconcavity of Throughput in Certain Closed Queueing Networks, in: Performance Evaluation, 6(1986), pp.293–305
Gordon, K.D., Dowdy, L.W.: The Impact of Certain Parameter Estimation Errors in Queueing Network Models, in: ACM Perf. Eval. Rev., 2(May 1980) pp.3–9
see chapter 6.1.2.2.2 for the different definitions of relative workloads
Further results of concavity and convexity in Closed Queueing Networks theory are shown in chapter 6.1.2.2.3
Fiacco, A.V., McCormick, G.B.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques, New York 1968, pp.19–20
Kobayashi, H., Gerla, M.: Optimal Routing in Closed Queueing Networks, in: ACM Trans. Comp. Syst., 1(1983)4, p.303
Kobayashi, H., Gerla, M.: Optimal Routing in Closed Queueing Networks, in: ACM Trans. Comp. Syst., 1(1983)4, p.303
Shalev-Oren, S., Seidmann, A., Schweitzer, P J.: Analysis of Flexible Manufacturing Systems with Priority Scheduling: PMVA, in: Annals of Operation Research, 3(1985), pp.115–139
see chapter 4.2.2
see chapter 6.1.2.2
see Appendix A
Buzen, J.P.: Computational Algorithms for Closed Queueing Networks with Exponential Servers, in: Comm. ACM, 16(1973)9, p.530
see chapter 4.2.2 and 15
for a description of this solution procedure refer to chapter 6.1.1.1
Stecke, K.E.: On the Nonconcavity of Throughput in Certain Closed Queueing Networks, in: Performance Evaluation, 6(1986), pp.296–297
see chapter 6.1.1.1
Bazaraa, M.S., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, New York 1979, p.148
see Stecke, K.E., Solbcre, J.J.: The Optimal Planning of Computerized Manufacturing Systems, School of Industrial Engineering, Perdue University, Report No.20, West Lafayette, Indiana 1981, p. 125
see chapter 6.1.2.2
Vinod, B., Solberg, J.J.: The optimal design of flexible manufacturing systems, in: IJPR, 23(1985)23, pp.1141–1151
see chapter 6.1.1.2
Vinod, B., Solberg, J.J.: The optimal design of flexible manufacturing systems, in: LJPR, 23(1985)23, pp. 1146–1150
Dallery, Y., Frein, Y.: An efficient Method to determine the optimal Configuration of a Flexible Manufacturing System, in: Proc. 2nd ORSA/TIMS Conf. on Flexible Manufacturing Systems: Operations Research Models and Applications, Ed.: K.E. Stecke and R. Suri, Amsterdam 1986, pp.272–275
A proof is given in Shanthikumar, J.G., Yao, D.D.: Optimal Server Allocation in a System of Multi-Server Stations, in: MS, 33(1987)9, pp.1173–1180
A proof is given in Suri, R.: A Concept of Monotonicity and Its Characterization for Closed Queueing Networks, in: OR, 33(1985)3, pp.606–624
Vinod, B., Solberg, J.J.: The optimal design of flexible manufacturing systems, in: LIPR, 23(1985)23, p. 1144
Vinod, B., Solberg, J.J.: The optimal design of flexible manufacturing systems, in: IJPR, 23(1985)23, p.1147
Dallery, Y., Frein, Y.: An efficient Method to determine the optimal Configuration of a Flexible Manufacturing System, in: Proc. 2nd ORSA/TIMS Conf. on Flexible Manufacturing Systems: Operations Research Models and Applications, Ed.: ICE. Stecke and R. Suri, Amsterdam 1986, p.2/0
Denning, P.J., Buzcn, J.P.: The operational analysis of qucueing network models, in: Computing Surveys, 10(1978)3, pp.225–262
Kleinrock, L. Queueing Systems, Vol.2 New York 1976 pp.219–225
see chapter 6.1.2.2
Dallery, Y., Frein, Y.: An Efficient Method to determine the Optimal Configuration of a Flexible Manufacturing System, in: Annals or OR, 15(1988), pp.207–225
for a description of mean value analysis see chapter 6.1.2.3.
Denning, P.J., Buzen, J.P.: The operational analysis of queueing network models, in: Computing Surveys, 10(1978)3, p.225
see chapter 6.1.2.2
Shanthikumar, J.G., Yao, D.D.: On server allocation in multiple center manufacturing systems, in: OR, 36(1988)2, pp.333–342;
Yao, D.D., Shanthikumar, J.G.: Some resource allocation problems in multi-cell systems, in: Proc. 2nd ORSA/TIMS Conf. on Flexible Manufacturing Systems: Operations Research Models and Applications, edited by K.E. Stecke and R. Suri, Amsterdam 1986, pp.249–250
A proof can be found in Shanthikumar, J.G., Yao, D.D.: On server allocation in multiple center manufacturing systems, in: OR, 36(1988)2, pp.335–336
Eager, D.L., Sevcik, K.C.: Performance Bound Hierarchies for Queueing Networks, in: ACM Trans. Comp. Syst., 1(1983), pp.99–115
Shanthikumar, J.G., Yao, D.D.: On server allocation in multiple center manufacturing systems, in: OR, 36(1988)2, p.338;
Fox, B.: Discrete Optimization via Marginal Analysis, in: MS, 13(1966), pp.210–216
see chapter 6.1.2.2
Shanthikumar, J.G., Yao, D.D.: Optimal server allocation in a system of multi-server stations, in: MS, 33(1987)9, p.1174
Fox, B.: Discrete Optimization via Marginal Analysis, in: MS, 13(1966), pp.210–216
Shanthikumar, J.G., Yao, D.D.: Optimal server allocation in a system of multi-server stations, in: MS, 33(1987)9, p.1176
see definitions in chapter 2.1
Yao, D.D., Shanthikumar, J.G.: Some resource allocation problems in multi-cell systems, in: Proc. 2nd ORSA/TIMS Conf. on Flexible Manufacturing Systems: Operations Research Models and Applications, edited by K.E. Stecke and R. Suri, Amsterdam 1986, pp.245–249;
Shanthikumar, J.G., Yao, D.D.: Optimal Buffer Allocation in a Multiceli System, in: The International Journal of Flexible Manufacturing Systems, 1(1989), pp.347–356
see chapter 6.1.2.2 and Gordon, W.J., Newell, G.F.: Closed Queueing Networks with Exponential Servers, in: OR, 15(1967), pp.252–267
Jackson, J.R.: Jobshop-Likc Queueing Systems, in: MS, 10(1963), pp.131–142
Fox, B.: Discrete Optimization via Marginal Analysis, in: MS, 13(1966), pp.210–216
Yao, D.D., Shanthikumar, J.G.: Some resource allocation problems in multi-cell systems, in: Proc. 2nd ORSA/TIMS Conf. on Flexible Manufacturing Systems: Operations Research Models and Applications, edited by K.E. Stecke and R. Suri, Amsterdam 1986, p.248
Note that in the original model formulation by Shanthikumar and Yao constraint (8.2.5.2) is an equality. Above the constraint has been changed to an inequality. This is because it is possible that all cells have been eliminated before the actual number of buffers reaches N.
Yao, D.D., Shanthikumar, J.G.: Some resource allocation problems in multi-cell systems, in: Proc. 2nd ORSA/TIMS Conf. on Flexible Manufacturing Systems: Operations Research Models and Applications, edited by ICE. Stecke and R. Suri, Amsterdam 1986, p.247
Jackson, J.R.: Jobshop-Like Queueing Systems, in: MS, 10(1963), pp.131–142;
Gordon, W.J., Newell, G.F.: Closed Queueing Networks with Exponential Servers, in: OR, 15(1967), pp.252–267
Solot, P.: Optimizing a Flexible Manufacturing System with several Pallet Types, Working Paper O.R.W.P. 87/17, Ecole Polytechnique Fédéral de Lausanne, Département de Mathématiques, Lausanne Sept. 1987, p.7
Solot, P.: Optimizing a Flexible Manufacturing System with several Pallet Types, Working Paper O.R.W.P. 87/17, Ecole Polytechnique Fédéral de Lausanne, Département de Mathématiques, Lausanne Sept. 1987, p.7
Solot, P.: Optimizing a Flexible Manufacturing System with several Pallet Types, Working Paper O.R.W.P. 87/17, Ecole Polytechnique Fédéral de Lausanne, Département de Mathématiques, Lausanne Sept. 1987, p.10
a description of the program MULTIQ can be found in: Solot, P., Bastos, J.M.: Choosing a Queueing Model for FMS, Working Paper O.R.W.P. 87/05, Ecole Polytechnique Fédéral de Lausanne, Département de Mathématiques, Lausanne 1987
see chapter 6.1.2.2
Zahorjan, J., Sevcik, K.C., Eager, D.L., Galler, B.: Balanced Job Bound Analysis of Queueing Networks, in: Comm. ACM, 25(1982)2, pp.134–141;
Eager, D.L., Sevcik, K.C.: Performance Bound Hierarchies for Queueing Networks, in: ACM Trans. Comp. Syst., 1(1983), pp.99–115
see chapter 6.1.2.2
Graves, S.C., Whitney, D.E.: A Mathematical Programming Procedure for Equipment Selection and System Evaluation in Programming Assembly, in: Proc. 18th IEEE Conf. on Decision and Control, Fort Lauderdale 1979, p.531
Graves, S.C., Whitney, D.E.: A Mathematical Programming Procedure for Equipment Selection and System Evaluation in Programming Assembly, in: Proc. 18th IEEE Conf. on Decision and Control, Fort Lauderdale 1979, S.532–533; Note that in the presented Lagrangian Function (6) the right hand side of the relaxed constraint is missing.
see chapter 7.5 for an alternative approach
Graves, S.C., Lamar, B.W.: A Mathematical Programming Procedure for Manufacturing System Design and Evaluation, in: Proc. IEEE Int. Conf. on Cir. and Comput., 1980, p.1147
Graves, S.C., Lamar, B.W.: A Mathematical Programming Procedure for Manufacturing System Design and Evaluation, in: Proc. IEEE Int. Conf. on Cir. and Comput., 1980, p.1147
for a more detailed description of the algorithm see: Graves, S.C., Lamar, B.W.: A Mathematical Programming Procedure for Manufacturing System Design and Evaluation, in: Proc. IEEE Int. Conf. on Cir. and Comput., 1980, p.1148–1149;
Graves, S.C., Lamar, B.W.: An Integer Programming Procedure for Assembly System Design Problems, in: OR, 31(1983)3, p.531–537
see chapter 7.5 for an alternative approach
see chapter for example chapter 8.1.1.1 or 8.1.1.2
Erlenkotter, D.: A comparative study of approaches to dynamic location problems, in: EJOR, 6(1981), p. 135
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Held, M., Wolfe, P., Crowder, H.D.: Validation of subgradient optimization, in: Mathematical Programming, 6(1974), pp.62–88
Fisher, M.L.: The Lagrangian Relaxation Method for solving Integer Programming Problems, in: Management Science, 27(1981)1, p.8
Fisher, M.L., Northup, W.D., Shapiro, J.F.: Using Duality to solve Discrete Optimization Problems: Theory and Computational Experience, in: Mathematical Programming Study, 3(1975), pp.76–79
Shapiro, J.F.: A Survey of Lagrangian Techniques for Discrete Optimization, in: Annals of Discrete Mathematics, 5(1979), pp.132–135
Geoffrion, A.M.: Lagrangian Relaxation for Integer Prgramming, in: Mathematical Programming Study, 2(1974), pp.82–114
For a description of greedy algorithms see chapter 6.1.1.3
The variable in square brackets [] is rounded off to the next higher integer value.
The variable in square brackets [] is rounded off to the next higher integer value.
see results in Appendix D
For a more in-depth discussion on this subject, see chapter 7.3
Jacobsen, S.K.: Heuristic solution to dynamic plant location problems, in: M. Roubens (Ed.), Advances in Operation Research, Amsterdam 1977, pp.207–211
Jacobsen, S.K.: Heuristic solution to dynamic plant location problems, in: M. Roubens (Ed.), Advances in Operation Research, Amsterdam 1977, p.209
Jacobsen, S.K.: Heuristic solution to dynamic plant location problems, in: M. Roubens (Ed.), Advances in Operation Research, Amsterdam 1977, p.208
For a discussion on the priciples of the algorithm see chapter 8.3.1.3.
The variable in square brackets [] is rounded to the next higher integer value.
For a detailed description on ABA-bounds see for example Zahorjan, J., Sevcik, K.C., Eager, D.L., Galler, B.: Balanced Job Bound Analysis of Queueing Networks, in: Comm. ACM, 25(1982)2, pp. 134–141
see chapter 8.3.1.3
Suri, R.: A Concept of Monotonicity and Its Characterization for Closed Queueing Networks, in: OR, 33(1985)3, pp.606–624; See also chapter 6.1.2.2.3 for an overview on the properties of the throughput function.
Shanthikumar, J.G., Yao, D.: Second-Order Properties of the Throughput of a Closed Queueing Network, in: Mathematics of OR, 13(1988)3, p.525; See also chapter 6.1.2.2.3 for an overview on the properties of the throughput function.
see chapter 6.1.2.2
Cost minimal flow fractions q* arc those fractions which arc cost minimal under the considerations of the given constraints.
Shanthikumar, J.G., Yao, D.: Second-Order Properties of the Throughput of a Closed Queueing Network, in: Mathematics of OR, 13(1988)3, p.525; See also chapter 6.1.2.2.3 for an overview on the properties of the throughput function.
Moore, C.G.: Network Models for Large-Scale Time-Sharing Systems, Technical Report No.71–1, Department of Industrial Engineering, University of Michigan, Ann Arbor 1971, pp.47–52
Bruell, S.C., Balbo, G.: Computational Algorithms for Closed Queueing Networks, New York and Oxford 1980, p.52
Lazowska, E.D., Zahorjan, J., Graham, G.S., Sevcik, K.C.: Quantitative System Performance Computer System Analysis Using Queueing Network Models, Englewood Cliffs 1984, p.53
Shanthikumar, J.G.: On the superiority of balanced load in a flexible manufacturing system, technical report, Department of IE & OR, Syracuse University, New York, 1982, p.4
Note that the objective function in the above model considers the average operating costs for one part instead of all operating costs. This simplification is due to the fact that the production rate Rmin is constant.
see chapter 6.1.2.2
Whitney, C.K., Suri, R.: Algorithms for Part and Machine Selection in Flexible Manufacturing Systems, in: Annals of OR, 3(1985), p.243
Whitney, CK., Suri, R.: Algorithms for Part and Machine Selection in Flexible Manufacturing Systems, in: Annals of OR, 3(1985), p.252
Whitney, C.K., Suri, R.: Algorithms for Part and Machine Selection in Flexible Manufacturing Systems, in: Annals of OR, 3(1985), p.246
Whitney, C.K., Gaul, T.S.: Sequential Decision Procedures for Batching and Balancing in Flexible Manufacturing Systems, in: Annals of OR, 3(1985), pp.301–316
for a more detailed description of the Algorithm see: Whitney, C.K., Suri, R.: Algorithms for Part and Machine Selection in Flexible Manufacturing Systems, in: Annals of OR, 3(1985), pp.257–260
Whitney, C.K., Gaul, T.S.: Sequential Decision Procedures for Batching and Balancing in Flexible Manufacturing Systems, in: Annals of OR, 3(1985), pp.301–316
Sarin, S.C., Chen, C.S.: A Mathematical Model for Manufacturing System Selection, in: Flexible Manufacturing Systems: Methods and Studies, Ed.: A. Kusiak, Elsevier Science Publishers B.V. (North Holland) 1986, p.102
King, J.R.: Machine-Component Group Formation in Group Technology, presented at the Vth Intern. Conf. on Production Research, Amsterdam Aug. 1979, in: OMEGA, 8(1980)2, pp.193–199
King, J.R., Nakornchai, V.: Machine-Component Group Formation in Group Technology: Review and Extension, in: UPR, 20(1982)2, pp.117–133
Sarin, S.C., Chen, CS.: A Mathematical Model for Manufacturing System Selection, in: Flexible Manufacturing Systems: Methods and Studies, Ed. A. Kusiak, Elsevier Science Publishers B.V. (North Holland) 1986, p. 105
see chapter 4.2.1
Sarin, S.C., Chen, CS.: A Mathematical Model for Manufacturing System Selection, in: Flexible Manufacturing Systems: Methods and Studies, Ed. A. Kusiak, Elsevier Science Publishers B.V. (North Holland) 1986, p.100
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Tetzlaff, U.A.W. (1990). Models for the optimal design of flexible manufacturing systems. In: Optimal Design of Flexible Manufacturing Systems. Contributions to Management Science. Physica, Heidelberg. https://doi.org/10.1007/978-3-642-50317-7_8
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