The Design of Manufacturing Systems Part 2 doc

to determine corresponding to the maximum violation of constraint (1.33) for a given LP relaxation solution. If such an inequality is found, it is appended to the current LP and the method progresses as before. The procedure for solving the separation problem for given solution is as follows. 1. Given , the current LP relaxation solution, calculate for each i and t. 2. Let ϭ arg max i,t . The lot sizing reformulation RP and the projection PP of P result in much larger problems than P because of additional variables and constraints, while providing tighter LP relaxations. The number of constraints in PP increases exponentially with the number of time periods. Computational results in [6] using branch and bound, indicate that for up to four time periods, RP and PP require smaller CPU times and fewer search nodes than P. However, for more than four time periods, P performs best and PP worst in terms of CPU times. This happens in spite of the fact that RP and PP require fewer search nodes than P. The increased CPU time for RP and PP can be attributed to the large number of additional constraints involved. However, when a constraint generation scheme is coupled with branch and bound for solving RP and PP, formulation RP outperforms P whereas PP outperforms both P and RP as the problem size increases. The success of the projection cutting plane approach is owing to the fact that only a small fraction of the projection constraints (1.33) is sufficient to significantly reduce the relaxation gap. Remarks Previously we discussed some of the solution strategies for model P. Bounding and integer cuts, strong cutting planes, and Benders decomposition were described as methods of direct solution of P. Two nonconventional reformulations of the same model, RP and PP, and their solution strategies were also discussed. In summary, we can make the following remarks. 1. Straightforward branch and bound strategy is computationally expensive for P because of the large number of feasible nodes that need to be examined. 2. For large-scale models of the form of P, the LP relaxation gap can be reduced by generating lower bounds, integer cuts, and strong cutting planes. A branch bound strategy with the relaxation gap reduced performs much better that Benders decomposition for these problems. 3. The lot sizing substructure embedded in the planning problem can be used to obtain nonconven- tional formulations. Model RP contains more variables and constraints than P. Yet, it possesses a tighter linear programming relaxation. 4. The large number of constraints in RP can be handled through a constraint generation scheme. 5. Projection of RP onto a lower dimensional space produces PP, a model with fewer variables but many more constraints. 6. The large number of constraints in PP can be handled through a cutting plane strategy along with branch and bound, in which only violated constraints are added. 7. Computational results indicate that, for small models, reformulation and projection models do not provide appreciable gains. i ء ,t ء () E ء W ء y ء ,,() ⌰ it ء W it ء ϭ Q i0 Ϫ U i ␶ t y it ء ␶ ϭ1 t Α Ϫ ␪ it ء E it ء ϭ U i ␶ t y it ء ␶ Յ tϪ ␴ i ␶ t ء 1if ␪ i ␶ t ء 0Յ 0if ␪ i ␶ t ء 0Ͼ    ␶ Յ tϭ ␨ it ء ␴ i ␶ t ء ␪ i ␶ t ء ␶ ϭ1 t Α ϭ i ء , t ء () ⌰ it ء ␨ it ء Ϫ() ϩ © 2001 by CRC Press LLC 2 Feature-Based Design in Integrated Manufacturing 2.1 Introduction 2.2 Definition of Features and Feature Taxonomies 2.3 Feature-Based Design Approaches 2.4 Automated Feature Recognition and CAD Representation 2.5 Feature-Based Design Applications 2.6 Research Issues in Feature-Based Manufacturing Architecture of the Feature-Based Design System • Feature Recognition Techniques for Complex Parts • Multiple Interpretation of Features • Incorporation of Tolerancing Information in the Feature Model • Feature Data Exchange Mechanisms • Feature Mapping • Feature Relations Taxonomy • Manufacturability Evaluation • Ranking of Redesign Alternatives • Product Design Optimization • Dimension-Driven Geometric Approach • Effects of Using Parallel NC Machines 2.7 Summary 2.1 Introduction The sequential engineering approach to product design and development typically treats design and manufacturing as isolated activities. In this approach, the design department designs an artifact and throws it “over the wall” to the manufacturing department without taking into consideration the man- ufacturing capabilities and limitations of the shop floor. The manufacturing department, in turn, studies the design from a manufacturability viewpoint and throws it back “over the wall” to the design department with a list of manufacturing concerns. Typically, the artifact drawings go back and forth between the two departments until, eventually, the drawings are approved for production. Obviously, this situation pro- longs the product realization time. Also, the cost of making design changes increases sharply with time. Owing to global competition, many manufacturing industries are under intense pressure to compress the product realization time and cost. These industries have realized that the sequential engineering approach should be discarded in favor of the Concurrent Engineering (CE) approach. The CE approach assumes that design and manufacturing activities are highly interdependent. It emphasizes that crucial manufacturing issues should be considered at the design stage in order to decrease the number of design iterations. Within the CE context, major research effort is being devoted in the development of seamless integrated engineering design and Venkat Allada University of Missouri-Rolla © 2001 by CRC Press LLC . 20 01 by CRC Press LLC 2 Feature-Based Design in Integrated Manufacturing 2. 1 Introduction 2. 2 Definition of Features and Feature Taxonomies 2. 3 Feature-Based Design Approaches 2. 4. product design and development typically treats design and manufacturing as isolated activities. In this approach, the design department designs an artifact and throws it “over the wall” to the manufacturing. back “over the wall” to the design department with a list of manufacturing concerns. Typically, the artifact drawings go back and forth between the two departments until, eventually, the drawings