Integer programming models for logical layout design of modular machining lines
Introduction
The work was carried out in conjunction with the European Project INTAS 03-51-5501. An industrial partner needed to develop a decision aided system to solve cost effectively the problem of designing machining lines. The analysis of the problem and literature brought us to the following conclusions: (1) the design problem is close to the assembly line balancing problem; (2) few methods exist to solve line balancing problems in machining/process environment; (3) as far as can be determined, there are not methods for line balancing with parallel operations at the same station (the considered industrial problem deals with the multi-spindle stations, so, the grouping operations for parallel execution in the same station is a crucial issue here).
Because of this issues, our motivations for this research are the following:
- 1.
To open a new area for applications of line balancing methods. Often line balancing methods are studied in the assembly environment. This problem is an opportunity to extend this approach to machining/process environment.
- 2.
To generalize SALBP when several operations can be executed in parallel at the same station. This is especially important because a large number of actual industrial problems cannot be solved with the standard SALBP methods due to the fact that they consider only sequential execution of operations.
- 3.
To broaden SALBP when operation costs depend on where the corresponding operation is executed and in which combination. The assumption of SALBP that cost of operations is the same for all stations and assignments is too restrictive for some industrial applications.
- 4.
To show how the available mathematical programming solvers (ILOG Cplex, XpressMP, etc.) can be used to solve the problem exactly and within an acceptable calculation time. This software is now accessible for both industrial and academic communities. However, if truly efficient solutions are sought then the pertinent mathematical modeling needs to be done.
Let us examine the core of our research project. Our interest focuses on machining transfer lines (Groover, 1987, Hitomi, 1996) which are often used in the automotive industry. These lines are required when there is a need for high machining precision and productivity (Dashchenko, 2003). Such lines consist of a series of machining workstations. Each workstation activates one or several spindle units which perform simultaneously a fixed set of operations. The production rate is insured by imposing the cycle time. The parts to be manufactured are moved further down the line on a common transfer mechanism (most often this is a conveyor). The downtime of one station causes the whole line to come to a halt and thus the output is reduced to zero. An illustration is given in Fig. 1. The line is composed of 3 workstations with the line flow diagonally from left bottom to top right.
The first workstation is equipped with two opposing spindle units: the left unit with two spindles and the right hand side only one. The second workstation is single unit station situated to the left of the conveyer. This unit consists of two spindles. Finally, the last station is composed of one horizontal spindle unit (on the left) and one vertical unit (on the right).
The activation of spindle units is simultaneous (in parallel) at workstations. Moreover, the execution of one spindle unit leads to a simultaneous execution of the corresponding operations. Thus, all operations performed at the same station are done simultaneously. These multiple operating tools activated in parallel increase the throughput relatively to the serial production systems (where the spindles are activated in sequence).
The modular machining lines are advantageous: maintenance and overhaul are easier, installation is rapid and reconfiguration becomes possible (Mehrabi et al., 2000, Koren et al., 1999). Moreover, this type of line is a very profitable system, reliability is higher and fewer spare parts are required. Usually, for these lines, the set of all available spindle units (standard modules) is known. A set of available modules is formed on the basis of: market analysis; previous units; complementary unit design studies.
Then, based on this knowledge, we study the logical line layout design. We assume that the average cost of one station and of each available spindle unit are known. Designing a logical layout corresponds in selecting of a subset of spindle units from the given set and grouping them into workstations to construct a line for machining a given product. An operation can usually be performed by different available spindle units. The aim is to choose the best one to perform each operation minimizing the total investment cost. The line investment cost is estimated as the sum of stations and spindle unit costs.
The rest of the paper is organized into five sections. In Section 2, related works are reported. In Section 3, two integer programs are presented. Section 4 explains how to improve the models by reducing their size. Section 5 deals with the experimental study. Finally, Section 6 states the conclusion.
Section snippets
Related works
This design problem can be seen as belonging to family of Line Balancing problems. The Simple Assembly Line Balancing Problem (SALBP) deals with elementary operations which are executed sequentially at workstations. We refer to Baybars (1986), Erel and Sarin (1998), Rekiek, Dolgui, Dechambre, and Bratcu (2002) and Scholl (1999) for comprehensive surveys. Much research has been accomplished to develop effective solutions to solve exactly the SALBP. They are often based on branch and bound
Notations and data
Notations as well as our specific assumptions are:
- •
N is the set of operations that have to be performed on each part (drilling, milling, boring, etc.).
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B = {br|br ⊂ N} corresponds to the set of all available multi-spindle units, where each unit is defined by the subset of operations it performs. A multi-spindle unit br is physically composed of one or several tools performing simultaneously corresponding operations. For the sake of simplicity, the term block is used henceforth to refer to a
Data checking
In this section, we suggest some pre-processing treatments to reduce the model size and computation time for optimization. For example, by looking at constraints, we can delete some inconsistent blocks. This inconsistency means they can’t belong to feasible solutions. This treatment is performed on the sets B, Dbs and Dos via finding the inconsistent elements and removing them.
- 1.
A block br can be removed from the set B if one of the following situations occurs:
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if the execution time of the block br
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Experimental study
We have used ILOG CONCERT 2.0 technology and solved test instances with ILOG CPLEX 9.0 with default parameters. The instances are generated randomly but with characteristics close to real industrial problems. The purpose of the experimental study is twofold: first to show the benefit achieved by a better formulation of the problem and second to observe the influence of three problem characteristics: (1) precedence graph density, (2) number of blocks in the set B and (3) number of operations in
Conclusion
These last years have seen a drastic improvement in computer aided design which reduces product development times. Nevertheless, the design of the manufacturing systems have not seen such advancement. Thus, the time to design/redesign the system and build it becomes critical. A rapid design of systems composed of modular components is a possible solution. Hence, efforts have to be made to present approaches for designing such systems.
Decisions related to the design of modular machining lines
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