Discrete-time model for two-machine one-buffer transfer lines with buffer bypass and two capacity levels

Abstract

This article deals with the analytical modeling of transfer lines consisting of two machines decoupled by one finite buffer. The innovative contribution of this work consists in representing a particular behavior that can be found in a number of industrial applications, such as in the ceramics and electronics industries. Specifically, the buffer significantly affects the line’s performance as, when it is accumulating or releasing material (i.e., when one machine is operational and the other machine is under repair), it forces the operational machine to slow down. Conversely, when both machines are operational they can work at a higher capacity since the buffer is bypassed. Thus, two levels for the machine capacity can be identified, based on the conditions of the machines and, consequently, the state of the buffer. The system is modeled as a discrete-time, discrete-state Markov process. The resulting two-Machine one-Buffer Model with Buffer Bypass is here called 2M-1B-BB model. The analytical solution of the model is obtained and mathematical expressions of the most important performance measures are provided. Finally, some numerical results are discussed.

Keywords:

• Transfer line
• discrete-time model
• buffer bypass
• two capacity levels

Appendix

Analytical solution for the 2M-1B BB Model

In Section 3.4 the following guess for the solution was proposed: (A.1) where (A.2)

The objective here is to determine the paramters X, Y₁, Y₂, and Y₃.

By plugging Equation (58(58) ) into Equations (28(28) ) to (31(31) ), the equations are satisfied for the following expressions of the parameters introduced above: (A.3) (A.4) (A.5) (A.6) where (A.7) (A.8) and (A.9) (A.10) (A.11) (A.12) where (A.13) (A.14) (A.15) (A.16) (A.17) (A.18)

Note that in Equations (A.1(A.1) ) to (A.18(A.18) ), p′_i has been substituted by ρp_i according to Equation (2(2) ), for i = 1, 2.

Since there two solutions for each parameter, the complete internal solution for the 2M-1B BB model is (A.19) where C₁ and C₂ are normalizing constants.

Note that if ρ = 1—i.e., the machines have a single capacity as occurs in the basic model without buffer bypassing—the above solution reduces to the one reported in Gershwin (2002) and, specifically, Y_3j = 1 for j = 1, 2.

Now we have to complete the solution by considering the boundary equations.

Let us consider the lower boundary first. By adding Equations (8(8) ), (9(9) ), (11(11) ), (12(12) ), we find that (A.20)

Since every term on the right-hand side of Equation (10(10) ) is of internal form, we can write (A.21) Thus, Equation (A.17(A.17) ) can be rewritten as (A.22) or (A.23) Note that if j = 2, X_jY_1j − Y_2j = 0. If e₁ ≠ e₂ and j = 1, then X_jY_1j − Y_2j ≠ 0. Therefore, if e₁ ≠ e₂, C₁ = 0.

As a consequence, the solution of Equation (A.16(A.16) ) can be simplified as follows: (A.24) or, by dropping the j subscript: (A.25) where X corresponds to X₂, Y₁ to Y₁₂, Y₂ to Y₂₂, and Y₃ to Y₃₂ as defined in Equations (A.6(A.6) )–(A.14(A.14) ).

In order to determine the remaining lower boundary probabilities, since the expression for p(1, 0, 1) is given by Equation (A.18(A.18) ), Equations (8(8) ), (11(11) ), and (9(9) ) are three equations into three unknowns, whose solutions are (A.26) (A.27) (A.28)

As regards the upper boundary, we note that, similar to p(1, 0, 1), p(N − 1, 1, 0) can be expressed according to the internal form. Thus, (A.29) so that Equations (14(14) ), (16(16) ), and (17(17) ) involve three unknowns. By solving the corresponding system we obtain (A.30) (A.31) (A.32)

Hence, Equations (33(33) ) to (49(49) ) in Section 3.4 are derived.

Additional information

Notes on contributors

Elisa Gebennini

Elisa Gebennini, Eng. Ph.D., received a master’s degree (with honors) in Management Engineering from the University of Modena and Reggio Emilia (Italy) in 2005. Then, she was engaged in the XXI edition of the Ph.D. course in Industrial Innovation Engineering at the University of Modena and Reggio Emilia, Italy, and obtained a Ph.D. degree in 2009 with a thesis entitled Supply Chain and Recovery Systems: Models and Practices. She was a visiting student at the Massachusetts Institute of Technology. She is currently a Research Fellow at the Department of Engineering Sciences and Methods (University of Modena and Reggio Emilia). Her research activity is mainly focused on manufacturing systems and logistics. In particular, the main research activities can be listed as follows: development of analytical models for the performance assessment of production systems; development of analytical and simulation models for the optimal design and analysis of automated storage and handling systems (e.g., automated guided vehicles); development of innovative mixed-integer programming optimization models for supply chain design and management (e.g., dynamic multistage location–allocation problems); and development of innovative models for the recovery network design and analysis with focus on WEEE (Waste Electrical and Electronic Equipment). She has contributed to several research projects funded by companies located in Emilia Romagna region.

Andrea Grassi

Andrea Grassi received a master’s degree (with honors) in Mechanical Engineering from the University of Parma, Italy. He was engaged in the XV edition of the Ph.D. Course in Production Systems and Industrial Plants at the University of Parma, Italy, and obtained a Ph.D. degree with a thesis concerning production scheduling. He is currently an Associate Professor at the Department of Engineering Sciences and Methods (University of Modena and Reggio Emilia). He was vice director of the Doctorate School in Industrial Innovation Engineering in the period 2011–2013. He is currently a member of the Direction Board of the “Associazione Italiana Docenti di Impiantistica”. His research activity is mainly focused on the solution of problems inherent manufacturing systems in their various aspects (production, logistics), among them, the development of heuristic algorithms for the solution of combinatorial problems, the development of stochastic analytical models for performance assessment, the development of methodologies to assess safety in manufacturing systems, the design of service equipment, the development of analytical models for logistics and for maintenance management, the optimization of production processes in food industry, and the development of analytical and simulative models for automated production line design. Results of his research activity have been published in international journals and in national and international conferences. He is responsible for various research projects funded by companies located in Emilia Romagna region and cooperates in research projects funded by multinational companies.